A weighted goal programming approach to estimate the linear regression model in full quasi type-2 fuzzy environment

Abstract

This study attempts to develop a quasi type-2 fuzzy regression model in full quasi type-2 fuzzy environment. To estimate the parameters of the proposed model, first, a weighted distance between quasi type-2 fuzzy numbers is defined based on L1-norm. Then some approximations for multiplication of two quasi type-2 fuzzy numbers (QT2FNs) are introduced. The problem of estimation of the parameters relies on a non-linear optimization problem, which is converted to a linear optimization problem. The method can handel both symmetric and asymmetric data. Two real world examples demonstrate the feasibility and efficiency of the proposed method. The predictive performance of the model is examined by cross-validation, and a similarity measure is used to compare our model with a similar model.

Keywords

Fuzzy linear regression goal programming type-2 fuzzy set uncertainty quasi type-2 fuzzy number

1 Introduction

Fuzzy set theory is a tool to handle the real world problems in which the information are uncertain and ambiguous. In such situations, a membership function is used which assigns a real number in [0, 1] for a membership degree of an element is a set. In many real-world problems the exact membership degree may not be identified. A fuzzy set with uncertainty in the membership function is called type-2 fuzzy set. Such sets can be used in situations where there are uncertainty in both membership degree and in the shape of the membership function.

Some examples of such situations have been identified in [17], e.g., when a measurement is corrupted with nonstationary noise and the mathematical description of nonstationary is unknown; when features in a pattern recognition application have statistical attributes that are nonstationary and the mathematical descriptions of the nonstationarities are unknown; when membership values are extracted from a group of experts using questionnaries, etc. Studies on Type-2 fuzzy sets briefy summarized as follows.

Karnik and Mendel [14], discussed in set theoretic operations for type-2 sets, properties of membership grades of type-2 sets, and type-2 relations and other compositions, and cartesian products under minimum and product t-norms. Mendel and John [17] defined a new representation theorem of type-2 fuzzy sets and introduced formulas for the union, intersection, and complement of type-2 fuzzy sets by using this new representation. Grzegorzewski [5] introduced some distances between interval-valued fuzzy sets based on Hausdorff metric. Hong and Lee [11] discussed some algebraic properties and a distance measure for interval-valued fuzzy numbers. Mendel and Liu [18] defined a quasi type-2 fuzzy logic system as a restricted special case of a type-2 fuzzy logic system represented by its α-planes.

Research on fuzzy regression analysis began by Celmin’s [2], Diamond [3] and Tanaka et al. [25], and was continued by some authors e.g. Wang and Tsaur [26]; Kao and Chyu [13]; Nasrabadi et al. [21]; Yao and Yu [28]; Hassanpour et al. [7 –10]; Kelkinnama and Taheri [15]. Recently, Rabiei et al. [23] proposed a Least-squares approach to regression modeling for interval-valued fuzzy data. The authors formulated a weighted goal programming model to calculate the quasi type-2 fuzzy coefficients of regression, when the input data are crisp and the output data are QT2FNs [12].

When there exist some outliers in the data set, it is usually preferred to use a robust approach. Traditionally, the regression analysis based on the method of least absolutes deviations was used as a robust method with respect to the least squares method in modeling a data set in which there were some outliers [4]. For example, Hassanpour et al. [8] proposed a goal programming approach based on least absolutes deviations to fuzzy regression modeling for triangular fuzzy input-output data. The effect of outliers is also compared with some methods in numerical examples. Their results showed that goal programming approach is less sensitive than the compared methods. In the present paper we introduce a goal programming model to estimate the regression coefficients when the regression coefficients, as well as the independent variables (inputs) and the response variable (output), are quasi type-2 fuzzy numbers. The rest of the paper is organized as follows: Section 2 contains some preliminaries of quasi type-2 fuzzy set theory. Also, a distance between quasi type-2 fuzzy numbers is introduced. In Section 3, we explain our proposed method. In Section 4, two real world data sets are used to illustrate how the proposed method is implemented. The predictive performance of the model is examined by cross-validation in Section 5. Finally, a brief conclusion is given in Section 6.

2 Preliminaries

Some concepts in type-2 fuzzy set theory are reviewed in this section [6 , 17]. Also, a weighted distance between two QT2FNs is introduced.

2.1 Type-2 fuzzy sets

A type-2 fuzzy set (T2FS), denoted by $\tilde{A}$ , in a crisp set X is characterised by a type-2 membership function $μ_{\tilde{A}} (x, u)$ , i.e., $\tilde{A} = {((x, u), μ_{\tilde{A}} (x, u)) | x \in X, u \in J_{x} \subseteq [0, 1]},$ where $0 \leq μ_{\tilde{A}} (x, u) \leq 1$ , u is a primary grade and $μ_{\tilde{A}} (x, u)$ is a secondary grade. A T2FS can be graphically shown in three dimensional space(3D). At each value of x, say x′, the two dimensional (2D) plane whose axes are u and $μ_{\tilde{A}} (x, u)$ is called a vertical slice (VS) of $\tilde{A}$ , i.e.,

$VS (x^{'}) = μ_{\tilde{A}} (x^{'}, u) \equiv μ_{\tilde{A}} (x^{'}) = {(u, f_{x^{'}} (u)) | u \in J_{x^{'}}}$

where f_x′ (u) : J_x → [0, 1] is a function that assigns a secondary grade to each primary grade u for some fixed x. The VS is a type-1 fuzzy set (T1FS) in [0, 1].

The Footprint Of Uncertainty (FOU) is derived from the union of all primary memberships. The FOU is bounded by two membership functions, a lower one, $\underline{μ_{\tilde{A}} (x)}$ and an upper one, $\bar{μ_{\tilde{A}} (x)}$ . The FOU can be described in terms of its upper and lower membership functions which themselves are T1FSs: $FOU (\tilde{A}) = {J_{x} | x \in X} = [\underline{FOU (\tilde{A})}, \bar{FOU (\tilde{A})}] .$ (1)

The principal membership function (PrMF) defined as the union of all the primary memberships having secondary grades equal to 1 $\Pr (\tilde{A}) = {(x, u) | x \in X, f_{x} (u) = 1} .$ (2)

An interval type-2 fuzzy set (IT2FS) is defined as a T2FS whose all secondary grades are of unity i.e. for all x, f_x (u) =1. An IT2FS can be completely determined by its FOU given by Equation (1).

Let $\tilde{A}$ be a T2FS satisfying the following propositions: [6]

A1: All the VSs of the T2FS are fuzzy numbers, i.e. $\forall x, h ({\tilde{A}}_{x}) = 1 .$

A2: All the VSs of the T2FS are piecewise functions of the same type (e.g. linear).

The first assumption assures that the T2FS contains an FOU and a Pr. This fact is clear since all the VSs are normal which makes it clear that for all the domain values there is at least one primary grade with secondary grade at unity. The second property assures that only a set parameters are needed to define this kind of T2FSs which is directly related to FOU and Pr. These assumptions allow this kind of T2FS be completely determined using its FOU and Pr, just like a T1FS which can be completely determined by its core and support, based on certain assumptions.

Definition 2.1. A T2FS is called a quasi type-2 fuzzy number (QT2FN) if it is completely determined by its FOU and Pr. The set of all QT2FNs is denoted by $QT 2 F (ℝ)$ .

The Extension Principle has been used by Zadeh [29] and Mizumoto and Tanaka [19] to derive the intersection and union of T2FSs. Karnik and Mendel [14] provide an in-depth investigation on these operations.

Theorem 2.1. [17] (QT2 Extension Principle) Let X = X₁ × ⋯ × X_n be the Cartesian product of universes, and ${\tilde{A}}_{1}, \dots, {\tilde{A}}_{n}$ be QT2FSs in each respective universe. Also let Y be another universe and $\tilde{B} \in Y$ be a QT2FS such that $\tilde{B} = f ({\tilde{A}}_{1}, \dots, {\tilde{A}}_{n})$ , where f : X ⟶ Y is a monotone mapping. Then application of Extension Principle to QT2FSs (QT2 Extension Principle) leads to the following: $\tilde{B} (y) = sup_{(x_{1}, \dots, x_{n}) \in f^{- 1} (y)} inf ({\tilde{A}}_{1} (x_{1}), \dots, {\tilde{A}}_{n} (x_{n}))$ where y = f (x₁, …, x_n).

Definition 2.2. Let $\tilde{A}$ and $\tilde{B}$ be two QT2FSs on the universal set X. Then, $\tilde{A}$ is called a subset of $\tilde{B}$ , denoted by $\tilde{A} \subseteq \tilde{B}$ , if for all x ∈ X, $\underline{A} (x) \leq \underline{B} (x), A (x) \leq B (x) and \bar{A} (x) \leq \bar{B} (x),$ where $\underline{A} (x) = \underline{FOU} (\tilde{A})$ , $A (x) = \Pr (\tilde{A})$ and $\bar{A} (x) = \bar{FOU} (\tilde{A})$ . In addition, $\tilde{A}$ is called equal to $\tilde{B}$ , denoted by $\tilde{A} = \tilde{B}$ , if for all x ∈ X, $\underline{A} (x) = \underline{B} (x) A (x) = B (x) and \bar{A} (x) = \bar{B} (x) .$

Definition 2.3. $\tilde{A} \in QT 2 F (ℝ)$ is called a positive QT2FN $(\tilde{A} > 0)$ , if $\bar{A} (x) = A (x) = \underline{A} (x) = 0$ whenever x ≤ 0; and $\tilde{A}$ is called a negative QT2FN $(\tilde{A} < 0)$ , if $\bar{A} (x) = A (x) = \underline{A} (x) = 0$ whenever x ≥ 0.

Definition 2.4. A QT2FN is called triangular if all of the membership functions $\underline{FOU} (\tilde{A})$ , $\bar{FOU} (\tilde{A})$ , $\Pr (\tilde{A})$ and vertical slices are triangular fuzzy numbers. A triangular QT2FN, say $\tilde{A}$ , is denoted by $\tilde{A} = (l_{1}, l, l_{2}, c, r_{2}, r, r_{1})$ . The parameters used here are $\Pr (\tilde{A}) = (l, c, r)$ and $FOU (\tilde{A}) = (l_{1}, l_{2}, c, r_{2}, r_{1})$ in which $(l_{1}, c, r_{1}) = \bar{FOU} (\tilde{A})$ and $(l_{2}, c, r_{2}) = \underline{FOU} (\tilde{A})$ .

The triangular QT2FN, $\tilde{A}$ can also be denoted by its left and right spreads as follows: $\tilde{A} = 〈 c; α_{1}, β_{1}; α, β; α_{2}, β_{2} 〉$ where α₁ = c - l₁, α = c - l, α₂ = c - l₂ and β₁ = r₁ - c, β = r - c, β₂ = r₂ - c are the left and right spreads of $\underline{FOU}$ , Pr and $\bar{FOU}$ , respectively, and 0 ≤ α₂ ≤ α ≤ α₁, 0 ≤ β₂ ≤ β ≤ β₁. (see Fig. 1).

$\tilde{A}$ is called symmetric if α₁ = β₁, α = β and α₂ = β₂. In such a case $\tilde{A}$ is denoted by $\tilde{A} = 〈 c; α_{1}; α; α_{2} 〉$ .

An ordinary triangular fuzzy number can be considered as a degenerated QT2FN in which all of the spreads of $\underline{FOU} (\tilde{A})$ , $\bar{FOU} (\tilde{A})$ and $\Pr (\tilde{A})$ are equal and their secondary membership functions are zero. Also, a real number can be considered as a degenerated QT2FN whose spreads and secondary membership functions are zero. The following formulas for addition of two triangular QT2FNs and multiplication of a triangular QT2FN by a scaler are drawn from extension principle of Hamrawi et al. [6].

Proposition 2.1. If $\tilde{A_{1}} = 〈 c; α_{1}, β_{1}; α, β; α_{2}, β_{2} 〉$ and $\tilde{A_{2}} = 〈 c^{'}; α_{1}^{'}, β_{1}^{'}; α^{'}, β^{'}; α_{2}^{'}, β_{2}^{'} 〉$ be in $QT 2 F (ℝ)$ and $λ \in ℝ$ , then $\begin{matrix} {\tilde{A}}_{1} \oplus {\tilde{A}}_{2} & = & 〈 c + c^{'}; α_{1} + α_{1}^{'}, β_{1} + β_{1}^{'}; \\ α + α^{'}, β + β^{'}; α_{2} + α_{2}^{'}, β_{2} + β_{2}^{'} 〉, \end{matrix}$ $λ {\tilde{A}}_{1} = {\begin{matrix} 〈 λ c; λ α_{1}, λ β_{1}; λ α, λ β; λ α_{2}, λ β_{2} 〉 & λ \geq 0, \\ 〈 λ c; - λ β_{1}, - λ α_{1}; - λ β, - λ α; - λ β_{2}, - λ α_{2} 〉 & λ < 0 . \end{matrix}$

For multiplication of two QT2FNs, there is not an analytically exact formula. The multiplication of two T2FNs based on extension principle is not necessarily a T2FN. Several multiplications are defined between type-1 fuzzy numbers, that their extension to T2FNs and using in field of fuzzy regression is needed more research. The following approximation can be considered in a similar way to that of Hassanpour et al. [8].

Definition 2.5. For two QT2FNs $\tilde{A_{1}}, \tilde{A_{2}}$ we define: $\begin{matrix} {\tilde{A}}_{1} \otimes {\tilde{A}}_{2} \\ ≃ {\begin{matrix} 〈 {cc}^{'}; c α_{1}^{'} + c^{'} α_{1}, c β_{1}^{'} + c^{'} β_{1}; c α^{'} + c^{'} α, c β^{'} + c^{'} β; \\ c α_{2}^{'} + c^{'} α_{2}, c β_{2}^{'} + c^{'} β_{2} 〉 & \tilde{A_{1}} > 0, \tilde{A_{2}} > 0 \\ 〈 {cc}^{'}; c^{'} α_{1} - c β_{1}^{'}, c^{'} β_{1} - c α_{1}^{'}; c^{'} α - c β^{'}, c^{'} β - c α^{'}; \\ c^{'} α_{2} - c β_{2}^{'}, c^{'} β_{2} - c α_{2}^{'} 〉 & \tilde{A_{1}} < 0, \tilde{A_{2}} > 0 \\ 〈 {cc}^{'}; - c^{'} β_{1} - c β_{1}^{'}, - c^{'} α_{1} - c α_{1}^{'}; - c^{'} β - c β^{'}, - c^{'} α - \\ c α^{'}; - c^{'} β_{2} - c β_{2}^{'}, - c^{'} α_{2} - c α_{2}^{'} 〉 & \tilde{A_{1}} < 0, \tilde{A_{2}} < 0 \end{matrix} \end{matrix}$ (3)

Proposition 2.2. Suppose $\tilde{A}$ , $\tilde{B}$ and $\tilde{C}$ be quasi type-2 fuzzy numbers. Then $(\tilde{A} \oplus \tilde{B}) \otimes \tilde{C} \subseteq (\tilde{A} \otimes \tilde{C}) \oplus (\tilde{B} \otimes \tilde{C}) .$ (4) The equality holds if $\tilde{A}, \tilde{B}, \tilde{C} > 0$ .

Proof. According to Definition 2.2, Relation (4) is satisfied if and only if this relation is satisfied for lower, principal and upper fuzzy numbers which define $\tilde{A}$ . In other words $\begin{matrix} (\tilde{A} \oplus \tilde{B}) \otimes \tilde{C} \subseteq (\tilde{A} \otimes \tilde{C}) \oplus (\tilde{B} \otimes \tilde{C}) \\ \Leftrightarrow {\begin{matrix} [\underline{A} (x) \oplus \underline{B} (x)] \otimes \underline{C} (x) \leq [\underline{A} (x) \otimes \underline{C} (x)] \oplus [\underline{B} (x) \otimes \underline{C} (x)], \\ [A (x) \oplus B (x)] \otimes C (x) \leq [A (x) \otimes C (x)] \oplus [B (x) \otimes C (x)], \\ [\bar{A} (x) \oplus \bar{B} (x)] \otimes \bar{C} (x) \leq [\bar{A} (x) \otimes \bar{C} (x)] \oplus [\bar{B} (x) \otimes \bar{C} (x)] . \end{matrix} \end{matrix}$

But, using a similar argument for fuzzy sets, the righthand side relations hold (see [9], Proposition 2.7), and the proof is complete. □

2.2 A distance between quasi type-2 fuzzy numbers

The observed data in our study are assumed to be QT2FN. So, we try to close the membership functions of observed and estimated responses from fuzzy linear regression model by closing their corresponding parameters. Since QT2FNs are completely characterized by the parameters of their FOU and Pr, closing the parameteres of two QTFNs is enough (in fact necessary and sufficient) to close their membership functions, which is the purpose of this paper. To do this, based on the distance defined between two triangular fuzzy numbers by Hassanpour et al. v, we propose the following weighted distance between QT2FNs in which different weights (w_i) are used to show different importance of the parameters [12].

Definition 2.6. [8] Let $\tilde{A} = 〈 c; α_{1}, β_{1}; α, β; α_{2}, β_{2} 〉$ and $\tilde{B} = 〈 c^{'}; α_{1}^{'}, β_{1}^{'}; α^{'}, β^{'}; α_{2}^{'}, β_{2}^{'} 〉$ be in $QT 2 F (ℝ)$ and w_i > 0 for i = 1, ⋯ , 7. Define the distance between $\tilde{A}$ and $\tilde{B}$ as follows: $\begin{matrix} d_{w} (\tilde{A}, \tilde{B}) = w_{1} | α_{1} - α_{1}^{'} | + w_{2} | β_{1} - β_{1}^{'} | + w_{3} | α_{2} - α_{2}^{'} | \\ + w_{4} | β_{2} - β_{2}^{'} | + w_{5} | α - α^{'} | + w_{6} | c - c^{'} | + w_{7} | β - β^{'} | . \end{matrix}$

Among advantages of using d_w, we can refer to its ease in both theory and application. Furthermore, the special formula of the proposed distance helps us to convert the nonlinear programming model proposed to calculate the regression coefficients, to a linear one. The advantages of this conversion are that solving linear programming (LP) problems is very easy, and their exact solution can be obtained by the Simplex method. However, most of available algorithms for solving nonlinear programming problems yield approximate solutions. In addition, depending on the relative importance of the parameters of quasi type-2 fuzzy numbers, we can assign different values to the weights w_i and obtain different solutions. For example, one can set w_i = 1 for all i, if the parameters of Pr and FOU have the same importance, and can set w₅ = w₆ = w₇ = 2, w₁ = w₂ = w₃ = w₄ = 1 if the importance of Pr is twice the importance of FOU.

Proposition 2.3. The function d_w defined in Definition 2.2 is a metric on $QT 2 F (ℝ) \times QT 2 F (ℝ)$ , i.e. for each $\tilde{A}, \tilde{B}, \tilde{C} \in QT 2 F (ℝ)$ we have:

$d_{w} (\tilde{A}, \tilde{B}) \geq 0$ and $d_{w} (\tilde{A}, \tilde{A}) = 0$ .

$d_{w} (\tilde{A}, \tilde{B}) = d_{w} (\tilde{B}, \tilde{A})$ .

$d_{w} (\tilde{A}, \tilde{C}) \leq d_{w} (\tilde{A}, \tilde{B}) + d_{w} (\tilde{B}, \tilde{C})$ .

Proof. Straightforward. □

Remark 2.1. In practice, we set $\sum_{i = 1}^{7} w_{i} = 1$ , to avoid repeated weights. Furtheremore, if we set w_i = 1 ∀ i, for two crisp numbers, the above distance is reduced to the absolute difference between them.

Remark 2.2. For two triangular fuzzy numbers, the above distance is reduced to the wighted sum of absolute difference between their middle point and their spreads (in fact, it is an extension of the metric introduced in [9] on T1FRs).

3 The proposed regression model

Consider a set of QT2F data ${({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, \dots, {\tilde{x}}_{ip},$ $\tilde{y_{i}}) | i = 1, \dots, n}$ , in which ${\tilde{x}}_{ij} (i = 1, \dots, n, j = 1, 2, \dots, p$ ) is the value of ith independent variable ( ${\tilde{x}}_{j}$ ) and ${\tilde{y}}_{i} (i = 1, \dots, n)$ is the corresponding value of dependent variable $\tilde{y}$ in the ith case. The purpose of quasi type-2 fuzzy linear regression (QT2FLR) is to fit a quasi type-2 fuzzy linear model to the given QT2F data. This model can be considered as follows: $\tilde{Y} = {\tilde{A}}_{0} \oplus ({\tilde{A}}_{1} \otimes {\tilde{x}}_{1}) \oplus ({\tilde{A}}_{2} \otimes {\tilde{x}}_{2}) \oplus \dots \oplus ({\tilde{A}}_{p} \otimes {\tilde{x}}_{p}) .$ (5)

In model (5), the coefficients ${\tilde{A}}_{0}, {\tilde{A}}_{1}, \dots, {\tilde{A}}_{p}$ are assumed to be QT2FNs. These parameters must be estimated such that for i = 1, …, n, the estimated responses ${\tilde{Y}}_{i}$ , ${\tilde{Y}}_{i} = {\tilde{A}}_{0} \oplus ({\tilde{A}}_{1} \otimes {\tilde{x}}_{i 1}) \oplus ({\tilde{A}}_{2} \otimes {\tilde{x}}_{i 2}) \oplus \dots \oplus ({\tilde{A}}_{p} \otimes {\tilde{x}}_{ip}),$ (6) be close to the corresponding fuzzy observed responses ${\tilde{y}}_{i} (i = 1, \dots, n)$ , as much as possible.

In this paper, the QT2FLR coefficients are supposed to be non-symmetric QT2FN. Also, given inputs ${\tilde{x}}_{ij} = 〈 c_{{\tilde{x}}_{ij}}$ ; $α_{1_{{\tilde{x}}_{ij}}}$ , $β_{1 {\tilde{x}}_{ij}}$ ; $α_{{\tilde{x}}_{ij}}$ , $β_{{\tilde{x}}_{ij}}$ ; $α_{2 {\tilde{x}}_{ij}}$ , $β_{2 {\tilde{x}}_{ij}} 〉$ , i = 1, …, n, j = 1, …, p, are supposed to be positive non-symmetric QT2FNs, by a simple translation of all data, if necessary. In addition, suppose that the observed responses are non-symmetric QT2FNs $\tilde{y_{i}} = 〈 c_{{\tilde{y}}_{i}}; α_{1 {\tilde{y}}_{i}}, β_{1 {\tilde{y}}_{i}}; α_{{\tilde{y}}_{i}}, β_{{\tilde{y}}_{i}}; α_{2 {\tilde{y}}_{i}}, β_{2 {\tilde{y}}_{i}} 〉$ , i = 1, …, n.

To calculate the regression coefficients, it is necessary to evaluate the product of two QT2FNs ${\tilde{A}}_{j}$ and ${\tilde{x}}_{ij}$ in (6). It is clear that the multiplication A ⊗ B in (2.1) depends on the sign of QT2FNs. Therefore, one must formulate different models for different states of the sign of regression coefficients. We attempt to formulate a non-linear programming (NLP) model for estimating the regression coefficients which are independent of the sign of regression coefficients. Another problem with using the multiplication (2.1) is that it is proposed only for two positive and/or negative QT2FNs. So, it cannot be used for QT2FNs whose supports contain both positive and negative real numbers. Therefore, we approximate the multiplication of two QT2FNs by some (in fact by infinite) QT2FNs. Then, the best approximation is chosen among them, to minimize a suitable function.

Note that a real number A has infinite representations in the form of A = A′ - A″, where A′ and A″ are nonnegative real numbers.

Suppose $\tilde{A} = 〈 c_{\tilde{A}}; α_{1 \tilde{A}}, β_{1 \tilde{A}}; α_{\tilde{A}}, β_{\tilde{A}}; α_{2 \tilde{A}}, β_{2 \tilde{A}} 〉$ and $\tilde{x} = 〈 c_{\tilde{x}}; α_{1 \tilde{x}}, β_{1 \tilde{x}}; α_{\tilde{x}}, β_{\tilde{x}}; α_{2 \tilde{x}}, β_{2 \tilde{x}} 〉$ are two QT2FNs where $\tilde{A}$ is unrestricted in sign and $\tilde{x}$ is a positive QT2FN. Set $\tilde{A} = \tilde{A^{'}} \oplus \tilde{A^{″}}$ in which $\begin{matrix} \tilde{A^{'}} & = & 〈 c_{\tilde{A^{'}}}; α_{1 \tilde{A^{'}}}, β_{1 \tilde{A^{'}}}; α_{\tilde{A^{'}}}, β_{\tilde{A^{'}}}; α_{2 \tilde{A^{'}}}, β_{2 \tilde{A^{'}}} 〉 \\ \tilde{A^{″}} & = & 〈 - c_{\tilde{A^{″}}}; α_{1 \tilde{A^{″}}}, β_{1 \tilde{A^{″}}}; α_{\tilde{A^{″}}}, β_{\tilde{A^{″}}}; α_{2 \tilde{A^{″}}}, β_{2 \tilde{A^{″}}} 〉 . \end{matrix}$ We have $\begin{matrix} c_{\tilde{A}} & = & c_{\tilde{A^{'}}} - c_{\tilde{A^{″}}}, α_{1 \tilde{A}} = α_{1 \tilde{A^{'}}} + α_{1 \tilde{A^{″}}}, β_{1 \tilde{A}} \\ = & β_{1 \tilde{A^{'}}} + β_{1 \tilde{A^{″}}}, α_{\tilde{A}} = α_{\tilde{A^{'}}} + α_{\tilde{A^{″}}}, \\ β_{\tilde{A}} & = & β_{\tilde{A^{'}}} + β_{\tilde{A^{″}}}, α_{2 \tilde{A}} = α_{2 \tilde{A^{'}}} + α_{2 \tilde{A^{″}}}, β_{2 \tilde{A}} = β_{2 \tilde{A^{'}}} + β_{2 \tilde{A^{″}}}, \\ α_{1 \tilde{A^{'}}}, α_{1 \tilde{A^{″}}}, β_{1 \tilde{A^{'}}}, β_{1 \tilde{A^{″}}}, α_{\tilde{A^{'}}}, α_{\tilde{A^{″}}}, β_{\tilde{A^{'}}}, β_{\tilde{A^{″}}}, α_{2 \tilde{A^{'}}}, \\ α_{2 \tilde{A^{″}}} & \geq & 0 . \end{matrix}$

By using Proposition 2.2, approximation (3), and addition (3) we have: $\begin{matrix} \tilde{A} \otimes \tilde{x} \\ = (\tilde{A^{'}} \oplus \tilde{A^{″}}) \otimes \tilde{x} \subseteq (\tilde{A^{'}} \otimes \tilde{x}) \oplus (\tilde{A^{″}} \otimes \tilde{x}) \\ ≃ 〈 c_{\tilde{A^{'}}} c_{\tilde{x}}; c_{\tilde{A^{'}}} α_{1 \tilde{x}} + c_{\tilde{x}} α_{1 \tilde{A^{'}}}, c_{\tilde{A^{'}}} β_{1 \tilde{x}} \\ + c_{\tilde{x}} β_{1 \tilde{A^{'}}}; c_{\tilde{A^{'}}} α_{\tilde{x}} + c_{\tilde{x}} α_{\tilde{A^{'}}}, c_{\tilde{A^{'}}} β_{\tilde{x}} \\ + c_{\tilde{x}} β_{\tilde{A^{'}}}; c_{\tilde{A^{'}}} α_{2 \tilde{x}} + c_{\tilde{x}} α_{2 \tilde{A^{'}}}, c_{\tilde{A^{'}}} β_{2 \tilde{x}} \\ + c_{\tilde{x}} β_{2 \tilde{A^{'}}} 〉 \oplus 〈 - c_{\tilde{A^{″}}} c_{\tilde{x}}; c_{\tilde{x}} α_{1 \tilde{A^{″}}} + c_{\tilde{A^{″}}} β_{1 \tilde{x}}, c_{\tilde{x}} β_{1 \tilde{A^{″}}} \\ + c_{\tilde{A^{″}}} α_{1 \tilde{x}}; c_{\tilde{x}} α_{\tilde{A^{″}}} + c_{\tilde{A^{″}}} β_{\tilde{x}}, c_{\tilde{x}} β_{\tilde{A^{″}}} \\ + c_{\tilde{A^{″}}} α_{\tilde{x}}; c_{\tilde{x}} α_{2 \tilde{A^{″}}} + c_{\tilde{A^{″}}} β_{2 \tilde{x}}, c_{\tilde{x}} β_{2 \tilde{A^{″}}} + c_{\tilde{A^{″}}} α_{2 \tilde{x}} 〉 \\ = 〈 (c_{\tilde{A^{'}}} - c_{\tilde{A^{″}}}) c_{\tilde{x}}; c_{\tilde{A^{'}}} α_{1 \tilde{x}} + c_{\tilde{x}} α_{1 \tilde{A^{'}}} + c_{\tilde{x}} α_{1 \tilde{A^{″}}} \\ + c_{\tilde{A^{″}}} β_{1 \tilde{x}}, c_{\tilde{A^{'}}} β_{1 \tilde{x}} + c_{\tilde{x}} β_{1 \tilde{A^{'}}} + c_{\tilde{x}} β_{1 \tilde{A^{″}}} \\ + c_{\tilde{A^{″}}} α_{1 \tilde{x}}; c_{\tilde{A^{'}}} α_{\tilde{x}} + c_{\tilde{x}} α_{\tilde{A^{'}}} + c_{\tilde{x}} α_{\tilde{A^{″}}} \\ + c_{\tilde{A^{″}}} β_{\tilde{x}}, c_{\tilde{A^{'}}} β_{\tilde{x}} + c_{\tilde{x}} β_{\tilde{A^{'}}} + c_{\tilde{x}} β_{\tilde{A^{″}}} \\ + c_{\tilde{A^{″}}} α_{\tilde{x}}; c_{\tilde{A^{'}}} α_{2 \tilde{x}} + c_{\tilde{x}} α_{2 \tilde{A^{'}}} + c_{\tilde{x}} α_{2 \tilde{A^{″}}} \\ + c_{\tilde{A^{″}}} β_{2 \tilde{x}}, c_{\tilde{A^{'}}} β_{2 \tilde{x}} + c_{\tilde{x}} β_{2 \tilde{A^{'}}} \\ + c_{\tilde{x}} β_{2 \tilde{A^{″}}} + c_{\tilde{A^{″}}} α_{2 \tilde{x}} 〉 \end{matrix}$ (7)

Let ${\tilde{A}}_{j} = \tilde{A_{j}^{'}} \oplus \tilde{A_{j}^{″}}$ for j = 1, …, p, where $\begin{matrix} \tilde{A_{j}^{'}} & = & 〈 c_{\tilde{A_{j}^{'}}}; α_{1 \tilde{A_{j}^{'}}}, β_{1 \tilde{A_{j}^{'}}}; α_{\tilde{A_{j}^{'}}}, β_{\tilde{A_{j}^{'}}}; α_{2 \tilde{A_{j}^{'}}}, β_{2 \tilde{A_{j}^{'}}} 〉 \\ \tilde{A ″_{j}} & = & 〈 - c_{\tilde{A_{j}^{″}}}; α_{1 \tilde{A_{j}^{″}}}, β_{1 \tilde{A^{″}}}; α_{\tilde{A_{j}^{″}}}, β_{\tilde{A_{j}^{″}}}; α_{2 \tilde{A_{j}^{″}}}, β_{2 \tilde{A_{j}^{″}}} 〉 \end{matrix}$ and $\tilde{A_{j}^{'}} > 0, \tilde{A_{j}^{″}} < 0$ . For each choice of $\tilde{A_{j}^{'}}$ and $\tilde{A_{j}^{″}}$ we have the following relation for ith estimated response ${\tilde{Y}}_{i}$ , i = 1, …, n $\begin{matrix} {\tilde{Y}}_{i} = \tilde{A_{0}} \oplus (\tilde{A_{1}} \otimes \tilde{x_{i 1}}) \oplus \dots \oplus (\tilde{A_{p}} \otimes \tilde{x_{ip}}) \\ = \tilde{A_{0}} \oplus [({\tilde{A_{1}}}^{'} \oplus {\tilde{A_{1}}}^{″}) \otimes \tilde{x_{i 1}}] \oplus \dots \oplus \\ [({\tilde{A_{p}}}^{'} \oplus {\tilde{A_{p}}}^{″}) \otimes \tilde{x_{ip}}] \\ \subseteq \tilde{A_{0}} \oplus [({\tilde{A_{1}}}^{'} \otimes \tilde{x_{i 1}}) \oplus ({\tilde{A_{1}}}^{″} \otimes \tilde{x_{i 1}})] \oplus \dots \oplus \\ [({\tilde{A_{p}}}^{'} \otimes \tilde{x_{ip}}) \oplus ({\tilde{A_{p}}}^{″} \otimes \tilde{x_{ip}})] \end{matrix}$ (8) We consider the right hand side of (8) as an approximation for ${\tilde{Y}}_{i}$ . Substituting $\tilde{A_{j}^{'}}$ and $\tilde{A_{j}^{″}}$ for j = 1, …, p in (8), by (3) and (3), we have

$\begin{matrix} {\tilde{Y}}_{i} ≃ 〈 c_{\tilde{A_{0}}} + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} - c_{\tilde{A_{j}^{″}}}) c_{\tilde{x_{ij}}}; α_{1_{\tilde{A_{0}}}} \\ + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} α_{1 \tilde{x_{ij}}} + c_{\tilde{x_{ij}}} α_{1 \tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} α_{1 \tilde{A_{j}^{″}}} + c_{\tilde{A_{j}^{″}}} β_{1 x_{ij}}), β_{1_{\tilde{A_{0}}}} \\ + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} β_{1 \tilde{x_{ij}}} + c_{\tilde{x_{ij}}} β_{1 \tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} β_{1 \tilde{A_{j}^{″}}} + c_{\tilde{A_{j}^{″}}} α_{1 \tilde{x_{ij}}}); α_{\tilde{a_{0}}} \\ + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} α_{\tilde{x_{ij}}} + c_{\tilde{x_{ij}}} α_{\tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} α_{\tilde{A_{j}^{″}}} + c_{\tilde{A_{j}^{″}}} β_{x_{ij}}), β_{\tilde{A_{0}}} \\ + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} β_{\tilde{x_{ij}}} + c_{\tilde{x_{ij}}} β_{\tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} β_{\tilde{A_{j}^{″}}} + c_{\tilde{A_{j}^{″}}} α_{\tilde{x_{ij}}}); α_{2_{\tilde{A_{0}}}} \\ + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} α_{2 \tilde{x_{ij}}} + c_{\tilde{x_{ij}}} α_{2 \tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} α_{2 \tilde{A_{j}^{″}}} + c_{\tilde{A_{j}^{″}}} β_{2 x_{ij}}), β_{2_{\tilde{A_{0}}}} \\ + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} β_{2 \tilde{x_{ij}}} + c_{\tilde{x_{ij}}} β_{2 \tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} β_{2 \tilde{A_{j}^{″}}} + c_{\tilde{A_{j}^{″}}} α_{2 \tilde{x_{ij}}}) 〉 \end{matrix}$ (9) Let us denote the right hand side of approximation (9) by ${\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})$ where $\tilde{A^{'}}$ and $\tilde{A^{″}}$ are p-dimensional vectors with jth element $\tilde{A_{j}^{'}}$ and $\tilde{A_{j}^{″}}$ , respectively. Clearly, for each choice of $\tilde{A^{'}}$ and $\tilde{A^{″}}$ , ${\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})$ is a triangular QT2FN, and we consider it as an approximation for ${\tilde{Y}}_{i}$ (Fig. 2). Therefore, there are many approximations for ${\tilde{Y}}_{i}$ , among which, we try to choose the best. To this end, we attempt to close the membership function of each approximated response ${\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})$ to that of corresponding observed response ${\tilde{y}}_{i}$ as much as possible. Therefore, we introduce the following mathematical programming model, which finds the best choices of $\tilde{A^{'}}$ and $\tilde{A^{″}}$ for the coefficients of model (6):

$min \sum_{i = 1}^{n} d_{w} ({\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}}), {\tilde{y}}_{i})$ (10) $\begin{matrix} s . t . & c_{\tilde{A_{0}}} \in ℝ, c_{\tilde{A_{j}^{'}}}, c_{\tilde{A_{j}^{″}}} \geq 0, j = 1, \dots, p, \\ α_{1 \tilde{A_{j}^{'}}}, β_{1 \tilde{A_{j}^{'}}}, α_{\tilde{A_{j}^{'}}}, β_{\tilde{A_{j}^{'}}}, α_{2 \tilde{A_{j}^{'}}}, β_{2 \tilde{A_{j}^{'}}} \geq 0, \\ α_{1 \tilde{A_{j}^{″}}}, β_{1 \tilde{A_{j}^{″}}}, α_{\tilde{A_{j}^{″}}}, β_{\tilde{A_{j}^{″}}}, α_{2 \tilde{A_{j}^{″}}}, β_{2 \tilde{A_{j}^{″}}} \geq 0, \\ 0 \leq α_{2} \leq α \leq α_{1}, 0 \leq β_{2} \leq β \leq β_{1} \end{matrix}$ where d_w was introduced in Definition 2.6. The model (10) can be converted to a GP model by choosing appropriate deviation variables. To this end, for i = 1 ⋯ n set

$\begin{matrix} \tilde{Y_{i}} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}}) = 〈 c_{{\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})}; α_{1 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})}, β_{1 {\tilde{Y}}_{i} (\tilde{A^{'}}, \tilde{A^{″}})}; \\ α_{{\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})}, β_{{\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})}; α_{2 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})}, β_{2 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} 〉 \end{matrix}$ and define: $\begin{matrix} n_{ik} = \frac{1}{2} {| k_{{\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} - k_{y_{i}} | - (k_{{\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} - k_{y_{i}})}, \\ p_{ik} = \frac{1}{2} {| k_{{\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} - k_{y_{i}} | + (k_{{\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} - k_{y_{i}})}, \end{matrix}$

for k = c, α₁, β₁, α, β, α₂, β₂. In fact, n_ik and p_ik (k ∈ {c, α₁, β₁, α, β, α₂, β₂}) are the negative and positive deviations between the parameters of the ith estimated and observed response, respectively. It can be easily seen that $\begin{matrix} n_{i α_{1}} & = & {\begin{matrix} α_{1 {\tilde{y}}_{i}} - α_{1 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} & α_{1 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} \leq α_{1 {\tilde{y}}_{i}}, \\ 0 & α_{1 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} > α_{1 {\tilde{y}}_{i}}, \end{matrix} \\ p_{i α_{1}} & = & {\begin{matrix} α_{1 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} - α_{1 {\tilde{y}}_{i}} & α_{1 {\tilde{y}}_{i}} \leq α_{1 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})}, \\ 0 & α_{1 {\tilde{y}}_{i}} > α_{1 {\tilde{Y}}_{i} ({\tilde{A}}_{0}, \tilde{A^{'}}, \tilde{A^{″}})} . \end{matrix} \end{matrix}$

Similar relations are hold true for the other deviation variables. By using the above deviation variables, the model (10) converts to the following GP model:

$\begin{matrix} (WGP) : min z = \sum_{i = 1}^{n} (w_{1} (n_{ic} + p_{ic}) + w_{2} (n_{i α_{1}} \\ + p_{i α_{1}}) + w_{3} (n_{i β_{1}} + p_{i β_{1}}) + w_{4} (n_{i α} + p_{i α}) + w_{5} (n_{i β} \\ + p_{i β}) + w_{6} (n_{i β_{2}} + p_{i β_{2}}) + w_{7} (n_{i α_{2}} + p_{i α_{2}})) \\ s . t . \\ c_{\tilde{A_{0}}} + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} - c_{\tilde{A_{j}^{″}}}) c_{\tilde{x_{ij}}} + n_{ic} - p_{ic} = c_{{\tilde{y}}_{i}}, \end{matrix}$ (11) $\begin{matrix} α_{1_{\tilde{A_{0}}}} + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} α_{1 \tilde{x_{ij}}} + c_{\tilde{x_{ij}}} α_{1 \tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} α_{1 \tilde{A_{j}^{″}}} \\ + c_{\tilde{A_{j}^{″}}} β_{1 x_{ij}}) + n_{i α_{1}} - p_{i α_{1}} = α_{1 {\tilde{y}}_{i}}, \end{matrix}$ (12) $\begin{matrix} β_{1_{\tilde{A_{0}}}} + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} β_{1 \tilde{x_{ij}}} + c_{\tilde{x_{ij}}} β_{1 \tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} β_{1 \tilde{A_{j}^{″}}} \\ + c_{\tilde{A_{j}^{″}}} α_{1 \tilde{x_{ij}}}) + n_{i β_{1}} - p_{i β_{1}} = β_{1 {\tilde{y}}_{i}} \end{matrix}$ (13) $\begin{matrix} α_{\tilde{A_{0}}} + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} α_{\tilde{x_{ij}}} + c_{\tilde{x_{ij}}} α_{\tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} α_{\tilde{A_{j}^{″}}} \\ + c_{\tilde{A_{j}^{″}}} β_{x_{ij}}) + n_{i α} - p_{i α} = α_{{\tilde{y}}_{i}} \end{matrix}$ (14) $\begin{matrix} β_{\tilde{a_{0}}} + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} β_{\tilde{x_{ij}}} + c_{\tilde{x_{ij}}} β_{\tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} β_{\tilde{A_{j}^{″}}} \\ + c_{\tilde{A_{j}^{″}}} α_{\tilde{x_{ij}}}) + n_{i β} - p_{i β} = β_{{\tilde{y}}_{i}} \end{matrix}$ (15) $\begin{matrix} α_{2_{\tilde{a_{0}}}} + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} α_{2 \tilde{x_{ij}}} + c_{\tilde{x_{ij}}} α_{2 \tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} α_{2 \tilde{A_{j}^{″}}} \\ + c_{\tilde{A_{j}^{″}}} β_{2 x_{ij}}) + n_{i α_{2}} - p_{i α_{2}} = α_{2 {\tilde{y}}_{i}} \end{matrix}$ (16) $\begin{matrix} β_{2_{\tilde{A_{0}}}} + \sum_{j = 1}^{p} (c_{\tilde{A_{j}^{'}}} β_{2 \tilde{x_{ij}}} + c_{\tilde{x_{ij}}} β_{2 \tilde{A_{j}^{'}}} + c_{\tilde{x_{ij}}} β_{2 \tilde{A_{j}^{″}}} \\ + c_{\tilde{A_{j}^{″}}} α_{2 \tilde{x_{ij}}}) + n_{i β_{2}} - p_{i β_{2}} = β_{2 {\tilde{y}}_{i}} \end{matrix}$ (17) $n_{ik} p_{ik} = 0, i = 1, \dots, n,$ (18) $\begin{matrix} n_{ik}, p_{ik} \geq 0, k = c, α_{1}, β_{1}, α, β, α_{2}, β_{2} \\ c_{\tilde{A_{j}^{'}}} \geq α_{1 \tilde{A_{j}^{'}}}, c_{\tilde{A_{j}^{″}}} \geq β_{1 \tilde{A_{j}^{″}}}, \\ α_{2 \tilde{A_{j}^{'}}} \leq α_{\tilde{A_{j}^{'}}} \leq α_{1 \tilde{A_{j}^{'}}}, α_{2 \tilde{A_{j}^{″}}} \leq α_{\tilde{A_{j}^{″}}} \leq α_{1 \tilde{A_{j}^{″}}}, \\ β_{2 \tilde{A_{j}^{'}}} \leq β_{\tilde{A_{j}^{'}}} \leq β_{1 \tilde{A_{j}^{'}}}, β_{2 \tilde{A_{j}^{″}}} \leq β_{\tilde{A_{j}^{″}}} \leq β_{1 \tilde{A_{j}^{″}}}, \\ α_{1 \tilde{A_{j}^{'}}}, β_{1 \tilde{A_{j}^{'}}}, α_{\tilde{A_{j}^{'}}}, β_{\tilde{A_{j}^{'}}}, α_{2 \tilde{A_{j}^{'}}}, β_{2 \tilde{A_{j}^{'}}} \geq 0, j = 1, \dots, p, \\ α_{1 \tilde{A_{j}^{″}}}, β_{1 \tilde{A_{j}^{″}}}, α_{\tilde{A_{j}^{″}}}, β_{\tilde{A_{j}^{″}}}, α_{2 \tilde{A_{j}^{″}}}, β_{2 \tilde{A_{j}^{″}}} \geq 0, j = 1, \dots, p \\ c_{\tilde{A_{0}}} \in ℝ, c_{\tilde{A_{j}^{'}}}, c_{\tilde{A_{j}^{″}}} \geq 0, j = 1, \dots, p . \end{matrix}$

It is clear that if $α_{{\tilde{y}}_{i}} = β_{{\tilde{y}}_{i}}$ for all i, we can set $α_{\tilde{x_{ij}}} = β_{\tilde{x_{ij}}}$ for each i and j. Accordingly, the constraints (14) and (15) will be equivalent, and one of them (in fact n constraints) can be removed. Similarly, if $α_{1 {\tilde{y}}_{i}} = β_{1 {\tilde{y}}_{i}}$ ( $α_{2 {\tilde{y}}_{i}} = β_{2 {\tilde{y}}_{i}}$ ) for all i, we can set $α_{1 \tilde{x_{ij}}} = β_{1 \tilde{x_{ij}}}$ ( $α_{1 {\tilde{x}}_{ij}} = β_{1 {\tilde{x}}_{ij}}$ ) for each i and j. Therefore, the constraints (12) and (13) ((16) and (17)) will be be equivalent, and the constraints (13) ((17)) can be removed, then we obtain a smaller model. In addition, one can remove the constraints (18) and solve the obtained LP model by the simplex method.

Since, this model finds the best approximation for ${\tilde{Y}}_{i}$ by minimizing the absolute deviations between the papameters of the observed and approximated responces, the problem always has feasible solutions (This is an advantage of goal programming problems).

The algorithm for constructing the QT2FLR model is as follows:

Step 1. Get p, n, and the input-output data $({\tilde{x}}_{i 1}, {\tilde{x}}_{i 2}, \dots, {\tilde{x}}_{ip}, {\tilde{y}}_{i})$ for i = 1, …, n.

Step 2. Solve the LP model obtained from WGP by removing the constraints (18) to obtain $c_{\tilde{A_{0}}}, α_{1 \tilde{A_{0}}}, β_{1 \tilde{A_{0}}}, α_{\tilde{A_{0}}}, β_{\tilde{A_{0}}}, α_{2 \tilde{A_{0}}}, β_{2 \tilde{A_{0}}}$ and $c_{\tilde{A_{j}^{'}}}, α_{1 \tilde{A_{j}^{'}}}, β_{1 \tilde{A_{j}^{'}}}, α_{\tilde{A_{j}^{'}}}, β_{\tilde{A_{j}^{'}}}, α_{2 \tilde{a_{j}^{'}}}, β_{2 \tilde{A_{j}^{'}}}, c_{\tilde{A_{j}^{″}}}, α_{1 \tilde{A_{j}^{″}}},$ $β_{1 \tilde{A_{j}^{″}}}, α_{\tilde{A_{j}^{″}}}, β_{\tilde{A_{j}^{″}}}, α_{2 \tilde{A_{j}^{″}}}, β_{2 \tilde{A_{j}^{″}}}$ for j = 1, …, p.

Step 3. Set ${\tilde{A}}_{0} = 〈 c_{\tilde{A_{0}}}; α_{1 \tilde{A_{0}}}, β_{1 \tilde{A_{0}}}; α_{\tilde{A_{0}}}, β_{\tilde{A_{0}}};$ $α_{2 \tilde{A_{0}}}, β_{2 \tilde{A_{0}}} 〉$ and calculate ${\tilde{A}}_{j} = {\tilde{A}}_{j}^{'} + {\tilde{A}}_{j}^{″}$ where ${\tilde{A}}_{j}^{'} = 〈 c_{\tilde{A_{j}^{'}}}; α_{1 \tilde{A_{j}^{'}}}, β_{1 \tilde{A_{j}^{'}}}; α_{\tilde{A_{j}^{'}}}, β_{\tilde{A_{j}^{'}}}; α_{2 \tilde{A_{j}^{'}}},$ $β_{2 \tilde{A_{j}^{'}}} 〉$ and ${\tilde{A}}_{j}^{″} = 〈 c_{\tilde{A_{j}^{″}}}; α_{1 \tilde{A_{j}^{″}}}, β_{1 \tilde{A_{j}^{″}}}; α_{\tilde{A_{j}^{″}}}, β_{\tilde{A_{j}^{″}}};$ $α_{2 \tilde{A_{j}^{″}}}, β_{2 \tilde{A_{j}^{″}}} 〉$ for j = 1, …, p. Put ${\tilde{A}}_{0}, \dots, {\tilde{A}}_{p}$ as the coefficients of the FLR model (5).

The complexity of simplex method for solving linear programming is exponential. One shortcoming of LP-based approaches is that the LP models which are used for estimating the regression coefficients have a large number of constraints, especially for a large number of observations. So the number of constraints increasing rapidly when the number of observations increases. Of course, with available LP solvers (e.g. MATLAB, LINGO and MPL), the size of LP is not important. Of course, there are other methods such as interior point methods which has polynomial time complexity.

Remark 3.1. The explained method can be applied to QT2FLR with triangular IT2FN outputs easily. Since a IT2FN is completely determined using its FOU, it is enough to modify the distance introduced in Definition 2.6 for two triangular IT2FNs $\tilde{A} = 〈 c; α_{1}, β_{1}; α_{2}, β_{2} 〉$ and $\tilde{B} = 〈 c^{'}; α_{1}^{'}, β_{1}^{'}; α_{2}^{'}, β_{2}^{'} 〉$ as follows: $\begin{matrix} d_{w} (\tilde{A}, \tilde{B}) & = & w_{1} | c - c^{'} | + w_{2} | α_{1} - α_{1}^{'} | + w_{3} | β_{1} - β_{1}^{'} | \\ + w_{4} | α_{2} - α_{2}^{'} | + w_{5} | β_{2} - β_{2}^{'} |, \end{matrix}$ (19) in which w₁, …, w₅ > 0. The WGP model in this case is obtained from (WGP) by removing the constraints (14) and (15).

4 Evaluation of the proposed regression model

In this section, we introduce two concepts for evaluating the proposed regression model.

4.1 Leave-one-out cross validation

To investigate the performance of the model, we apply an index based on the cross validation method [27] to examine the predictive ability of the models. To this end, for i = 1, …, n, the ith observation is left out from the data set, while the remaining observations are used to develop a quasi type-2 fuzzy regression model. Then the obtained model is used to predict the response value of the ith observation (denoted by ${\tilde{Y}}_{(- i)} (x_{i})$ ). Finally, to compare the ith observed response ${\tilde{y}}_{i}$ and the predicted value ${\tilde{Y}}_{(- i)} (x_{i})$ , we calculate the mean of distances d_w between y_i and ${\tilde{Y}}_{(- i)} (x_{i})$ which we call it MDC.

Definition 4.1. For QT2F regression model (5), the MDC index is defined by

$\begin{matrix} MDC = \frac{1}{n} \sum_{i = 1}^{n} d_{w} ({\tilde{Y}}_{(- i)} (x_{i}), {\tilde{y}}_{i}), \end{matrix}$ where ${\tilde{Y}}_{(- i)} (x_{i})$ is the QT2F response predicted by omitting the ith observation or the ith input-output data.

Definition 4.2. For QT2F regresion model (5), the mean of distances between estimated and observed values is defined by

$\begin{matrix} {MD}^{*} = \frac{1}{n} \sum_{i = 1}^{n} d_{w} ({\tilde{Y}}_{i}, {\tilde{y}}_{i}) \end{matrix}$

Now, by using the above indices, the relative error of the estimated responses can be calculated as $\begin{matrix} RE = \frac{| MDC - {MD}^{*} |}{{MD}^{*}} . \end{matrix}$

4.2 Goodness of fit of the IT2F regression model

To evaluate the goodness of fit of the interval type-2 fuzzy linear regression (IT2FLR) model, we use a weighted similarity measure between two IT2FNs ([1]), and it is used to compare our model with similar model.

Definition 4.3. [1] Let $\tilde{A}$ and $\tilde{B}$ be two IT2Fs of the universal set $X = ℝ$ . The weighted similarity measure between $\tilde{A}$ and $\tilde{B}$ is defined as follows: $\begin{matrix} {SM}_{w} (\tilde{A}, \tilde{B}) & = & 1 - w . \frac{\int_{\infty}^{- \infty} | \underline{μ_{\tilde{A}}} (x) - \underline{μ_{\tilde{B}}} (x) | dx}{\int_{\infty}^{- \infty} \underline{μ_{\tilde{A}}} (x) dx + \int_{\infty}^{- \infty} \underline{μ_{\tilde{B}}} (x) dx} \\ - (1 - w) . \frac{\int_{\infty}^{- \infty} | \bar{μ_{\tilde{A}}} (x) - \bar{μ_{\tilde{B}}} (x) | dx}{\int_{\infty}^{- \infty} \bar{μ_{\tilde{A}}} (x) dx + \int_{\infty}^{- \infty} \bar{μ_{\tilde{B}}} (x) dx} \end{matrix}$ (20) where $\underline{μ_{\tilde{A}}}$ and $\bar{μ_{\tilde{A}}}$ are the membership functions of $\underline{FOU} (\tilde{A})$ and $\bar{FOU} (\tilde{A})$ , respectively.

Proposition 4.1. [1] The mapping SM_w on the set of all IT2FNs $(IT 2 F (ℝ))$ satisfies the following properties: $\begin{matrix} i) {SM}_{w} (\tilde{A}, \tilde{B}) \in [0, 1], \\ ii) {SM}_{w} (\tilde{A}, \tilde{B}) = 1 \Leftrightarrow \tilde{A} = \tilde{B}, \\ iii) {SM}_{w} (\tilde{A}, \tilde{B}) = {SM}_{w} (\tilde{B}, \tilde{A}), \\ iv) \tilde{A} \subseteq \tilde{B} \subseteq \tilde{C} \Rightarrow {SM}_{w} (\tilde{A}, \tilde{C}) \leq {SM}_{w} (\tilde{A}, \tilde{B}), \\ {SM}_{w} (\tilde{A}, \tilde{C}) \leq {SM}_{w} (\tilde{B}, \tilde{C}) . \end{matrix}$

Remark 4.1. In Definition 4.3, it is obvious that the selection of w is more or less subjective, and it depends on the users opinion. The greater value of w means the more importance of $\underline{FOU}$ .

Definition 4.4. To evaluate goodness of ift of IT2F regression model, the maen of similarity measures between the observed values $\tilde{y_{i}}$ and the estimated values $\tilde{Y_{i}}$ , is defined as

$\bar{SM} = \frac{1}{n} \sum_{i = 1}^{n} {SM}_{w} (\tilde{y_{i}}, \tilde{Y_{i}})$

5 Application in soil science

In this section, we provide two applied examples to explain how the proposed method is applicable to derive regression model for quasi type-2 fuzzy observations.

One of the classical problems in soil science is the measurement of physical, chemical, and biological soil properties. The problem results from the difficulty, time and cost of direct measurements. Pedomodels, which have become a popular topic in soil science and environmental research, are predictive functions of certain soil properties based on other easily or cheaply measured properties [22]. Here, two pedomodels including one and two independent variables are studied to develop the relationships between different chemical and physical soil properties by means of quasi type-2 fuzzy regression technique. Based on a study in a part of Silakhor plain (situated in the province of Lorestan, west of Iran), different soil physical and chemical properties were measured using standard procedures. But, due to some impreciseness in experimental environment, the observed data were reported as IT2FNs [24]. Since examples containing quasi type-2 fuzzy data did not exist in previous works, we changed its output data to triangular QT2FNs.

5.1 Pedomodel of ESP-SAR

We first wish to provide a relationship between exchangeable sodium percentage (ESP), as the dependent variable, and sodium absorption ratio (SAR), as an independent variable. The exchange sodium percentage, ESP, governs the source/sink phenomenon for ionic constituents, i.e., sodium, as a contaminant in sodic soils, is calculated from the ratio of exchangeable sodium, Na_x, to cation exchangeable capacity, CEC. All these soil parameters, measured on soil colloidal surface, are time consuming and costly. In this case, ESP is considered as cost and time variable, therefore the need for less expensive indirect measurement is emphasized.

In this study, for each choice of ${\tilde{A}}_{1}^{'}$ and ${\tilde{A}}_{1}^{″}$ , the regression model for the data of Table 1 is as follows:

$\begin{matrix} \tilde{Y} & = & {\tilde{A}}_{0} \oplus ({\tilde{A}}_{1} \otimes {\tilde{x}}_{1}) \\ = & {\tilde{A}}_{0} \oplus ({\tilde{A}}_{1}^{'} \oplus {\tilde{A}}_{1}^{″}) \otimes {\tilde{x}}_{1} \end{matrix}$ (21)

In the above model, non-symmetric QT2FNs $\tilde{Y}$ and ${\tilde{x}}_{1}$ are cation sodium absorption ratio (SAR) and exchange sodium percentage (ESP), respectively.

According to the proposed method, the regression coefficients are obtained as $\begin{matrix} {\tilde{A}}_{0} & = & 〈 1.04; 0.01, 0; 0.01, 0; 0, 0 〉 \\ \tilde{A_{1}^{'}} & = & 〈 6.58; 0.01, 0.59; 0, 0.59; 0, 0.54 〉 \\ \tilde{A_{1}^{″}} & = & 〈 0.69; 0.01, 0.27; 0, 0.27; 0, 0.22 〉 . \end{matrix}$ using MATLAB software. The above QT2F regression model can be used to predict the ESP of a new case. For example, if for a new case, SAR = 〈1.08; 0.87, 0.91; 0.59, 0.64; 0.32, 0.38〉 then by Eq. (21), we predict the ESP as $\tilde{Y} = 〈 7.41; 6.41, 7.54; 4.35, 5.56; 2.37, 3.55 〉$ . The primary membership functions of $\tilde{Y}$ and the secondary membership function for x = 10 are shown in Fig. 3. Here, interpretation of the predicted value of ESP is done possibility. For instance, the possibility that ESP would be equal to 10 is a type-1 fuzzy number in which the center of this number has the maximum possibility and its lower and upper points have the minimum possibility. Note that, this type of interpretation to predict the value of ESP is based on possibility and consistency, that is principally different from the interpretation of statistical confidence intervals which is based on the concept of probability and relative frequency.

5.2 Pedomodel of CEC-OM-SAND

The second model provides a relationship between cation exchange capacity (CEC), as a function of two soil variables namely percentage of sand content (SAND) and organic matter content (OM). In the soil, organic matter can enhance the CEC, while the sand content has negative effect on the cation exchange capacity [20]. In this case, CEC is considered as cost and time variable, therefore the need for less expensive indirect measurement is emphasized.

In this study, for each choice of $\tilde{A_{j}^{'}}$ and $\tilde{A_{j}^{″}}$ , j = 1, 2 the regression model for the data of Table 3 is as follows

$\begin{matrix} \tilde{Y} = \tilde{A_{0}} \oplus (\tilde{A_{1}} \otimes {\tilde{x}}_{1}) \oplus (\tilde{A_{2}} \otimes {\tilde{x}}_{2}) \\ \tilde{A_{0}} \oplus (\tilde{A_{1}^{'}} \oplus \tilde{A_{1}^{″}}) \otimes {\tilde{x}}_{1} \oplus (\tilde{A_{2}^{'}} \oplus \tilde{A_{2}^{″}}) \otimes {\tilde{x}}_{2} \end{matrix}$ (22)

According to the proposed method, the estimated coefficients are obtained as $\begin{matrix} {\tilde{A}}_{0} & = & 〈 21.96; 13.49, 13.18; 8.19, 6; 5.48, 0.0 〉 \\ \tilde{A_{1}} & = & 〈 - 0.16; 0.0, 0.01; 0.0, 0.01; 0.0, 0.0 〉 \\ \tilde{A_{2}} & = & 〈 1.57; 4.93, 6.83; 4.93, 6.83; 3.11, 6.38 〉 . \end{matrix}$

In the above model, asymmetric QT2FNs $\tilde{y}$ , $\tilde{x_{1}}$ and $\tilde{x_{2}}$ are cation exchange capacity (CEC), percentage of sand content (SAND) and organic matter content (OM), respectively.

The above QT2F regression model can be used to predict the CEC of a new case. For example, if for a new case, $\begin{matrix} SAND = 〈 38; 36.2, 17.29; 30.04, 12.26; 23.88, 7.24 〉 \\ OM = 〈 0.84; 0.24, 0.35; 0.23, 0.29; 0.23, 0.23 〉 \end{matrix}$ then by Equation (22), the predicted CEC is $\tilde{Y} = 〈 17.01; 9.58, 14.79; 5.62, 8.67; 2.7, 2.7 〉 .$

In order to evaluate the predictive ability of the above models, the MDC is calculated for each model.

The value of the MDC for the ESP-SAR regression was obtained to be 19.9, and the value of MD^* is 14.25. Note that the relative error between MDC and MD^* is RE = 0.28.

The value of the MDC for the CEC-OM-SAND regression was obtained to be 24.55, which is close to the value of MD^*, i.e. 20.51. Note that the relative error between MDC and MD^* is RE = 0.16.

It is appeared that predictive ability to the MSL-DW model is much better than the CEC-OM-SAND model, and predictive ability to the CEC-SAND-OM model is better than the ESP-SAR model.

Recently, Rabiei et al. [23] used a distance on the space of interval type-2 fuzzy numbers (which determinated using its FOU) and proposed a least-squares method (LS) to obtain coefficients of the proposed model. In soil science examples, if input and output variables are considered to be interval-valued fuzzy numbers, we use the distance introduced in Eq. (19). In the case of interval type-2 fuzzy input-output data, the similarity measures between observed and predicted values in two regression model are shown in Table 2.

In soil science examples, the obtained results from comparison the maen of similarity mesaures, shows that the ${\bar{SM}}_{w} (GP)$ is less than ${\bar{SM}}_{w} (LS)$ , therefore, the observed and estimated values in proposed approach is more fit than the least squares model provided by rabiei et al. [24].

6 Conclusion

In some real systems, especially environmental systems, the relation between variables can be investigated in quasi type-2 fuzzy environment. In this work, we proposed a weighted goal programming approach to estimate the coefficients of quasi type-2 fuzzy linear regression model with quasi type-2 fuzzy input-output data and coefficients. The nonlinear programming model which has been presented to calculate the regression coefficient is converted to a goal programming model and then to a linear programmnig model easily. The advantage of this conversion is that linear programming problems can be solved exactly by available algorithms. Whereas, the available algorithms for solving nonlinear programming problems often give approximate solutions. The applicability of the proposed approach was investigated by using two real data sets in soil science. The predict ability of the model evaluated by cross-validation method. The proposed model is general and can be applied to any field of research. In soil science examples, the obtained results from comparison the maen of similarity mesaures, showed that, in the proposed goal programming approach, the observed and estimated values are more consistent than least squares model provided by rabiei et al.

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