The solution for type-2 fuzzy linear programming model based on the nearest interval approximation

Abstract

Linear programming is an important branch of operations research. The model is mature in theory and widely used in real life. However, various complex realistic scenarios involve fuzzy information. In this paper, we consider a fuzzy linear programming (FLP) model in which all parameters are trapezoidal interval type-2 fuzzy numbers (IT2FNs) and propose a solution method based on the nearest interval approximation and the best-worst cases (BWC) method. We prove the nearest interval approximation interval of trapezoidal IT2FNs, then the trapezoidal IT2FNs in the model are transformed into interval numbers which both upper and lower limits are interval numbers. With the help of best-worst cases (BWC) method, the sub-models of the transformed interval linear programming model are proposed, and four sub-solutions with different specific meanings can be obtained by solving them respectively. Finally, an application example is presented to show the rationality and practical significance of the method.

Keywords

Fuzzy linear programming (FLP)Trapezoidal interval type-2 fuzzy numbers (IT2FNs)Best-worst cases (BWC) method The nearest interval approximation

1 Introduction

Linear programming model first be described to solve the problem of cutting materials and production and transportation [1]. Dantzig formally proposed linear programming as a mathematical model and gave a simplex algorithm for its solution in 1947 [2]. Since then, linear programming has become more and more mature in theory and widely used.

However, when dealing with real problems, a lot of information is uncertain. Zadeh [3] first proposed the fuzzy idea in 1965. The emergence of fuzzy theory provides an effective tool for dealing with uncertain information in real life. With the increasing vagueness and uncertainty of evaluation information provided by experts, Type-1 Fuzzy Set is insufficient to address the subjective evaluation information with high complexity. Thus, Zadeh [4] proposed the concept of type-2 fuzzy in 1975, and Mendel [5] defined type-2 fuzzy set (T2FS). T2FS can better transfer the uncertainty of membership function [6], to depict more uncertainty in real life, especially the linguistic uncertainty. By comparison, Rough sets do not require prior information and have a lot of information loss [7]. Also, the picture fuzzy set may be adequate in situations when human opinions involving more answers of the type: yes, abstain, no, refusal [8]. Nevertheless, it is difficult to deal with and study general type-2 fuzzy numbers, so scholars put forward the concept of interval type-2 fuzzy number (IT2FNs) [5]. At present, most of the research in the field of type-2 fuzzy focuses on IT2FNs.

Since the effectiveness of fuzzy theory in modeling the uncertain and fuzzy preference information, it is widely used to solve decision problems. Wu et al. [9] proposed a new fuzzy decision-making framework based on Choquet integral and T2FSs for solving the problem of electric vehicle charging supplier selection. Abdullah and Najib [10] proposed a new fuzzy AHP method based on IT2FSs and utilized the proposed method to evaluate workplace safety. Tang et al. [11] combined the IT2FSs with the Fine-Kinney framework to analyze the risk of ballast tank maintenance. In general, as an effective tool for handling uncertainty, the fuzzy theory is widely applied to improve various methods.

Fuzzy linear programming (FTP) problem is an important application of fuzzy theory, which have many uncertain variations, with different optimization methods [12]. FTP is applied in many fields. For example, Ilbahar et al. [13] used a fuzzy linear programming model to investigate the most suitable locations for waste-to energy power plants in Central Anatolia Region of Turkey. Roy and Jana [14] analyzed the multi-objective linear production planning problem in triangular hesitant fuzzy environment. Lima et al. [15] used a mixed-integer linear programming model to address the strategic and tactical planning of a downstream oil supply chain subject to different sources ofuncertainty.

Many scholars have studied the FTP model with IT2FNs. Chance constraint programming (CCP) is one of the important methods to solve fuzzy parameter constraint optimization problems [16]. Kundu et al. [17] proposed a CCP method for solving linear programming models with IT2FNs constraints. Qin et al. [18] proposed a method to solve the data envelopment analysis (DEA) model with type-2 fuzzy coefficients by using the CV type-2 fuzzy variable reduction method. Golpayegani et al. [19] presented a new way in linear programming problems with special IT2FNs. Figueroa [20] enhanced a new method to solve fuzzy linear programming model with uncertain constraints based on α-cut.

To solve FTP problems, the common method is to reduce the fuzzy number to real number, and then to solve the real number linear programming model [21]. Many scholars put forward the ranking method of T2FS, such as Yager Index Rank [22], Centroid of IT2FSs [23], Extended Alpha Cuts [24] and so on. However, using ranking and other methods to reduce type-2 fuzzy numbers will lose more important information. It is a good compromise to transform fuzzy numbers into interval numbers. Grzegorzewski [25] discussed the interval approximation problem, proposed an nearest interval approximation method of type-1 fuzzy numbers, and proved that the proposed interval approximation is the best interval about a certain distance between fuzzy numbers.

After transforming fuzzy number into interval number, the model of interval linear programming problem can be considered. Da et al. [26] proposed a solution method based on fuzzy constraint satisfaction. Guo et al. [27] gave an interval number order relation reflecting the satisfaction of decision makers. On this basis, the interval linear programming is transformed into a deterministic linear programming. Tong [28] developed a best and worst cases (BWC) method to deal with interval linear programming model, that is, the linear programming model is transformed into two sub-models, representing the best and worst cases. Allahdadi et al. [29] improved and developed it. Chinneck et al. [30] extended BWC method to linear programming model with equal sign constraint.

To sum up, the existing works mainly focus on FTP models with partial parameters and use ranking methods. Trapezoidal IT2FNs is a kind of IT2FNs which can represent more uncertainties in real life and are closer to the way people present language. However, there is a certain gap in the study of linear programming models in which all parameters are trapezoidal IT2FNs.

In order to fill in this gap, we focus on this problem and propose a method for solving the FLP model in which all parameters are trapezoidal IT2FNs. We extend the nearest interval approximation of type-1 fuzzy numbers and prove the nearest interval approximation interval of trapezoidal IT2FNs. With the nearest interval approximation interval of trapezoidal IT2FNs, the numbers in the model are transformed into interval numbers which both upper and lower limits are interval numbers. With the help of BWC method, the transformed interval programming problem is transformed into four real linear programming models, which represent four different good and bad situations. Finally, an application example is presented to prove the rationality and practical significance of the method.

The proposed method provides a new perspective for the study of type-2 fuzzy linear programming model. The solution process is clear and easy to understand. At present, linear programming model has been widely used in industry, agriculture, business and other fields. This method can be used when parameters involve uncertain information, especially linguistic uncertainty.

The novelties of this article are shown in the following three points:

Trapezoidal interval type-2 fuzzy numbers can express more uncertainty information. Hovever, there is a certain gap in the research of linear programming model in which all parameters are trapezoidal interval type-2 fuzzy numbers.

It is more common to use sorting methods to transform fuzzy numbers into crisp values. However, it will lose more important information. So it is a better compromise choice to convert the fuzzy number into an interval and we extend the nearest interval approximation to trapezoidal IT2FNs.

The BWC method is a common method to solve interval linear programming. We use this method to solve transformed models.

The rest of this paper is organized as follows: Section 2 contains basic definitions and notations. Section 3 proves the nearest interval approximation interval of trapezoidal IT2FNs and proposes a method for solving the FLP model whose parameters are trapezoidal IT2FNs. Section 4 gives an application example to prove the rationality and practical significance of the proposed method. Section 5 is the conclusion of this paper.

2 Preliminaries

In this section, we review necessary concepts related to the fuzzy set theory and fuzzy linear programming models which will be used in the rest of this paper.

2.1 Basic concepts of Type-2 fuzzy set

Definition 1 [5] Let X be the universe of discourse, a type-2 fuzzy set (T2FS) $\tilde{A}$ can be represented by type-2 membership function $μ_{\tilde{A}} (x, u)$ as follows: $\tilde{A} = {(x, u), μ_{\tilde{A}} (x, u) | \forall x \in X, u \in Jx, 0 ⩽ μ_{\tilde{A}} (x, u) ⩽ 1}$ (1)

Where Jx denotes an interval in [0, 1]. Moreover, the T2FS $\tilde{A}$ can also be expressed as the following form: $\begin{matrix} \tilde{A} = \int_{x \in X} \int_{u \in Jx} μ_{\tilde{A}} (x, u) / (x, u) \\ = \int_{x \in X} (\int_{u \in Jx} μ_{\tilde{A}} (x, u) / (x, u)) / x \end{matrix}$ (2) Where Jx ⊆ [0, 1] is the primary membership at x, and $\int_{u \in Jx} μ_{\tilde{A}} (x, u) / (x, u)$ indicates the second membership at x. ∫∫ denotes the union over all admissible x and u. For discrete space, ∫ is replaced by ∑.

Definition 2 [5] Let $\tilde{A}$ be a T2FS in the universe of discourse X represented by type-2 membership function $μ_{\tilde{A}} (x, u)$ . If all $μ_{\tilde{A}} (x, u) = 1$ , then $\tilde{A}$ is called an interval type-2 fuzzy set (IT2FS). An interval type-2 fuzzy set can be regarded as a special case of the T2FS, which is defined as follows: $\tilde{A} = \int_{x \in X} \int_{u \in Jx} 1 / (x, u) = \int_{x \in X} (\int_{u \in Jx} 1 / (x, u)) / x$ (3)

Definition 3 [31 –33] Uncertainty in the primary memberships of T2FS $\tilde{A}$ consists of a bounded region called the footprint of uncertainty footprint (FOU), which may be expressed as follows: $FOU (\tilde{A}) = ⋃_{x \in X} x \times Jx$ (4)

The FOU of $\tilde{A}$ is bounded by two MFs: a lower membership function (LMF) $μ_{\tilde{A}}^{L} (x)$ and an upper membership function (UMF) $μ_{\tilde{A}}^{U} (x)$ . Thus, an IT2FS is bounded by two type-1 fuzzy sets. Moreover, the IT2FS is called the IT2FN when its UMF and LMF are type-1 fuzzy numbers. It is noteworthy that $\tilde{A}$ has embedded type-1 fuzzy sets (Ae) as well; therefore, there is an infinite amount of Ae enclosed into the FOU of $\tilde{A}$ .

Definition 4 [34] When the IT2FS is $\tilde{A} = ((a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; H_{1} ({\tilde{A}}^{L}), H_{2} ({\tilde{A}}^{L})), (a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U};$ $H_{1} ({\tilde{A}}^{U}), H_{2} ({\tilde{A}}^{U})))$ , the UMF is $μ_{\tilde{A}}^{U} (x)$ and the LMF is $μ_{\tilde{A}}^{L} (x)$ . The following formulas are given: $y = μ_{\tilde{A}}^{L} (x) = {\begin{matrix} H_{1} ({\tilde{A}}^{L}) \frac{x - a_{1}^{L}}{a_{2}^{L} - a_{1}^{L}}, a_{1}^{L} ⩽ x ⩽ a_{2}^{L} \\ [H_{2} ({\tilde{A}}^{L}) - H_{1} ({\tilde{A}}^{L})] \frac{x - a_{2}^{L}}{a_{3}^{L} - a_{2}^{L}}, a_{2}^{L} < x ⩽ a_{3}^{L} \\ H_{2} ({\tilde{A}}^{L}) \frac{a_{4}^{L} - x}{a_{4}^{L} - a_{3}^{L}}, a_{3}^{L} < x ⩽ a_{4}^{L} \\ 0, x < a_{1}^{L}, x > a_{4}^{L} \end{matrix}$ (5) $y = μ_{\tilde{A}}^{U} (x) = {\begin{matrix} H_{1} ({\tilde{A}}^{U}) \frac{x - a_{1}^{U}}{a_{2}^{U} - a_{1}^{U}}, a_{1}^{U} ⩽ x ⩽ a_{2}^{U} \\ [H_{2} ({\tilde{A}}^{U}) - H_{1} ({\tilde{A}}^{U})] \frac{x - a_{2}^{U}}{a_{3}^{U} - a_{2}^{U}}, a_{2}^{U} < x ⩽ a_{3}^{U} \\ H_{2} ({\tilde{A}}^{U}) \frac{a_{4}^{U} - x}{a_{4}^{U} - a_{3}^{U}}, a_{3}^{U} < x ⩽ a_{4}^{U} \\ 0, x < a_{1}^{U}, x > a_{4}^{U} \end{matrix}$ (6)

When $H_{1} ({\tilde{A}}^{L}) = H_{2} ({\tilde{A}}^{L}) = h_{\tilde{A}}^{L}$ , $H_{1} ({\tilde{A}}^{U}) = H_{2} ({\tilde{A}}^{U}) = h_{\tilde{A}}^{U}$ , it is a trapezoidal interval type-2 fuzzy numbers, $\tilde{A} = ((a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h_{\tilde{A}}^{L}), (a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h_{\tilde{A}}^{U}))$ . And the corresponding upper and lower membership functions are: $y = μ_{\tilde{A}}^{L} (x) = {\begin{matrix} h_{\tilde{A}}^{L} \frac{x - a_{1}^{L}}{a_{2}^{L} - a_{1}^{L}}, a_{1}^{L} ⩽ x ⩽ a_{2}^{L} \\ h_{\tilde{A}}^{L}, a_{2}^{L} < x ⩽ a_{3}^{L} \\ h_{\tilde{A}}^{L} \frac{a_{4}^{L} - x}{a_{4}^{L} - a_{3}^{L}}, a_{3}^{L} < x ⩽ a_{4}^{L} \\ 0, x < a_{1}^{L}, x > a_{4}^{L} \end{matrix}$ (7) $y = μ_{\tilde{A}}^{U} (x) = {\begin{matrix} h_{\tilde{A}}^{U} \frac{x - a_{1}^{U}}{a_{2}^{U} - a_{1}^{U}}, a_{1}^{U} ⩽ x ⩽ a_{2}^{U} \\ h_{\tilde{A}}^{U}, a_{2}^{U} < x ⩽ a_{3}^{U} \\ h_{\tilde{A}}^{U} \frac{a_{4}^{U} - x}{a_{4}^{U} - a_{3}^{U}}, a_{3}^{U} < x ⩽ a_{4}^{U} \\ 0, x < a_{1}^{U}, x > a_{4}^{U} \end{matrix}$ (8)

As shown below:

Fig. 1

Trapezoidal IT2FS.

2.2 The nearest interval approximation

Definition 5 [4] According to the Zadeh extension principle, the fuzzy set $\tilde{A}$ can be represented by its interval: $\tilde{A} = ⋃_{α} α \cdot A_{α}, α \in [0, 1]$ (9)

Where ${\tilde{A}}_{α}$ is the α-cut of fuzzy numbers $\tilde{A}$ $\begin{matrix} {\tilde{A}}_{α} = {x \in \tilde{X} | μ_{\tilde{A}} (x) ⩾ α} \\ = [min {x \in \tilde{X} | μ_{\tilde{A}} (x) ⩾ α}, max {x \in \tilde{X} | μ_{\tilde{A}} (x) ⩾ α}] \\ = [A_{α}^{L}, A_{α}^{U}] \end{matrix}$ (10)

Definition 6 [35] The α-cut of IT2FS $\tilde{A}$ are presented as follows: ${\tilde{A}}_{α} = {x | μ_{\tilde{A}}^{L} (x) ⩾ α, μ_{\tilde{A}}^{U} (x) ⩾ α} = [a (α), b (α)]$ where $a (α) \in {\begin{matrix} [a_{l} (α), a_{r} (α)], α \in [0, h_{\tilde{A}}^{L}] \\ [a_{l} (α), b_{r} (α)], α \in [h_{\tilde{A}}^{L}, h_{\tilde{A}}^{U}] \end{matrix},$ $b (α) \in {\begin{matrix} [b_{l} (α), b_{r} (α)], α \in [0, h_{\tilde{A}}^{L}] \\ [a_{l} (α), b_{r} (α)], α \in [h_{\tilde{A}}^{L}, h_{\tilde{A}}^{U}] \end{matrix}$

Fig. 2

α-cut of Trapezoidal IT2FS.

Definition 7 [25] Suppose A is a type-1 fuzzy number and $[\underline{A} (α), \bar{A} (α)]$ is its α-cut. The closed interval number $[\underline{C}, \bar{C}]$ is the nearest interval approximation to A, where $\underline{C} = \int_{0}^{1} \underline{A} (α) d α$ , $\bar{C} = \int_{0}^{1} \bar{A} (α) d α$ . When it is a triangular fuzzy number A = (a₁, a₂, a₃), its nearest interval approximation is $[\underline{C}, \bar{C}] = [\frac{a_{1} + a_{2}}{2}, \frac{a_{2} + a_{3}}{2}]$ . When it is a trapezoidal fuzzy number A = (a₁, a₂, a₃, a₄), its nearest interval approximation is $[\underline{C}, \bar{C}] = [\frac{a_{1} + a_{2}}{2}, \frac{a_{3} + a_{4}}{2}]$ .

2.3 Best-worst cases method

Definition 8 [28, 29] In the general Interval linear programming(ILP) problem, all the parameters (the objective function, technological coefficients and the resource values) are closed intervals. Therefore, the ILP problem can be modeled as follows: $\begin{matrix} max Z = \sum [{\underline{c}}_{j}, {\bar{c}}_{j}] x_{j} \\ s . t . \sum [{\underline{a}}_{ij}, {\bar{a}}_{ij}] x_{j} ⩽ [{\underline{b}}_{i}, {\bar{b}}_{j}], i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n, \end{matrix}$ (11)

Where ${\underline{c}}_{j}, {\bar{c}}_{j}, {\underline{a}}_{ij}, {\bar{a}}_{ij}, {\underline{b}}_{i}$ and ${\bar{b}}_{j}$ are real numbers.

BWC method solves the above model with the conversion of the ILP problem into the two sub-models, namely the best and the worst sub-models.

The best sub-model: $max {Z | \sum {\bar{a}}_{ij} x_{j} ⩽ {\underline{b}}_{i}, \forall x_{j} ⩾ 0}$ (12)

The worst sub-model: $max {Z | \sum {\bar{a}}_{ij} x_{j} ⩽ {\underline{b}}_{i}, \forall x_{j} ⩾ 0}$ (13)

Tong [28] proposed the above the best and worst Cases (BWC) method to solve the interval linear programming problem, and Allahdadi et al. [29] improved and developed it.

Type-2 fuzzy linear programming models classifified into two groups: fuzzy constraints and fuzzy cost parameters.

Figueroa-Garcìa and Hernández [20] used interval optimization approaches to solve linear programming problems with interval type-2 fuzzy constraints. An uncertain constrained model can be expressed as follows: $\begin{matrix} Max z = c^{t} x \\ s . t . Ax \tilde{⩽} \tilde{b} \\ x ⩾ 0 \end{matrix}$ (14)

Where x, c ∈ R^m, A ∈ R^n×m, $\tilde{b}$ is a type-2 fuzzy vector defined by two primary membership functions and $\tilde{⩽}$ is a Type-2 fuzzy partial order.

A model with type-2 fuzzy cost parameters in objective function is also considered in transportation model [36]. A type-2 fuzzy cost parameters model is expressed as follows: $\begin{matrix} Max z = {\tilde{c}}^{t} x \\ s . t . Ax ⩽ \tilde{b} \\ x ⩾ 0 \end{matrix}$ (15)

Where x ∈ R^m, and $\tilde{c}, \tilde{b}$ are type-2 fuzzy vectors.

In this paper, considering the more complex reality, all parameters in the linear programming model are regarded as type-2 fuzzynumbers.

3 Linear programming model based on nearest interval approximation

In this section, we extend the concept of the nearest interval approximation to trapezoidal IT2FNs. Based on this, a method for solving linear programming problem of trapezoidal IT2FNs is proposed by using BWC method.

3.1 The Nearest interval approximation of trapezoidal IT2FNs

We propose the nearest interval approximation of trapezoidal IT2FNs by using the proof idea of Grzegorzewski [25].

Theorem 1. Suppose $\tilde{A} = ((a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L}),$ $(a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}))$ is a trapezoidal ITFN. Its nearest interval approximation is $C_{d} (\tilde{A}) = [C_{L}, C_{R}]$ , where C_L ∈ [C_Ll, C_Lr] , C_U ∈ [C_Rl, C_Rr], $C_{Ll} = \frac{\int_{0}^{h^{U}} a_{l} (α) d α}{h^{U}} = a_{1}^{U} + \frac{(a_{2}^{U} - a_{1}^{U})}{2},$ $C_{Lr} = \frac{\int_{0}^{h^{L}} a_{r} (α) d α}{h^{L}} = a_{1}^{L} + \frac{(a_{2}^{L} - a_{1}^{L})}{2},$ $C_{Rl} = \frac{\int_{0}^{h^{L}} b_{l} (α) d α}{h^{L}} = a_{4}^{L} - \frac{(a_{4}^{L} - a_{3}^{L})}{2},$ $C_{Rr} = \frac{\int_{0}^{h^{U}} b_{r} (α) d α}{h^{U}} = a_{4}^{U} - \frac{(a_{4}^{U} - a_{3}^{U})}{2} .$

Proof:

Suppose $\tilde{A} = ((a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L}), (a_{1}^{U}, a_{2}^{U}, a_{3}^{U},$ $a_{4}^{U}; h^{U}))$ is a trapezoidal T2FN and [a (α) , b (α)] is its α-cut. Given $\tilde{A}$ we will try to find a closed interval $C_{d} (\tilde{A})$ which is the nearest to $\tilde{A}$ . We can do it since each interval is also a fuzzy number with constant -cuts for all α ∈ (0, 1]. Let $C_{d} (\tilde{A}) = [C_{L}, C_{R}]$ , where C_L ∈ [C_Ll, C_Lr] , C_U ∈ [C_Rl, C_Rr].

Assuming that $d (\tilde{A}, C_{d} (\tilde{A}))$ represents the distance between the fuzzy number $\tilde{A}$ and the interval approximation: $d (\tilde{A}, C (\tilde{A})) = \sqrt{\int_{0}^{h^{U}} \begin{matrix} (a_{l} (α) - C_{Ll})^{2} d α + \int_{0}^{h^{L}} (a_{r} (α) - C_{Lr})^{2} d α + \\ \int_{0}^{h^{L}} (b_{l} (α) - C_{Rl})^{2} d α + \int_{0}^{h^{U}} (b_{r} (α) - C_{Rr})^{2} d α \end{matrix}}$

In order to minimize the above function easily, we make $D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr}) = d^{2} (\tilde{A}, C (\tilde{A}))$ and find the partial derivative of this function: $\begin{matrix} \frac{\partial D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr})}{\partial C_{Ll}} = - 2 \int_{0}^{h^{U}} a_{l} (α) d α + 2 h^{U} C_{Ll} \\ \frac{\partial D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr})}{\partial C_{Lr}} = - 2 \int_{0}^{h^{L}} a_{r} (α) d α + 2 h^{L} C_{Lr} \\ \frac{\partial D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr})}{\partial C_{Rl}} = - 2 \int_{0}^{h^{L}} b_{l} (α) d α + 2 h^{L} C_{Rl} \\ \frac{\partial D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr})}{\partial C_{Rr}} = - 2 \int_{0}^{h^{U}} b_{r} (α) d α + 2 h^{U} C_{Rr} \end{matrix}$

Let ${\begin{matrix} \frac{\partial D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr})}{\partial C_{Ll}} = 0 \\ \frac{\partial D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr})}{\partial C_{Lr}} = 0 \\ \frac{\partial D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr})}{\partial C_{Rl}} = 0 \\ \frac{\partial D (C_{Ll}, C_{Lr}, C_{Rl}, C_{Rr})}{\partial C_{Rr}} = 0 \end{matrix}$

We have ${\begin{matrix} C_{Ll} = \frac{\int_{0}^{h^{U}} a_{l} (α) d α}{h^{U}} \\ C_{Lr} = \frac{\int_{0}^{h^{L}} a_{r} (α) d α}{h^{L}} \\ C_{Rl} = \frac{\int_{0}^{h^{L}} b_{l} (α) d α}{h^{L}} \\ C_{Rr} = \frac{\int_{0}^{h^{U}} \underline{b_{r}} (α) d α}{h^{U}} \end{matrix}$

As for $\tilde{A} = ((a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L}), (a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}))$ , its α-cut is [a (α) , b (α)], $a (α) \in {\begin{matrix} [a_{l} (α), a_{r} (α)] = [a_{1}^{U} + \frac{α (a_{2}^{U} - a_{1}^{U})}{h^{U}}, a_{1}^{L} + \frac{α (a_{2}^{L} - a_{1}^{L})}{h^{L}}], α \in [0, h^{L}] \\ [a_{l} (α), b_{r} (α)] = [a_{1}^{U} + \frac{α (a_{2}^{U} - a_{1}^{U})}{h^{U}}, a_{4}^{U} - \frac{α (a_{4}^{U} - a_{3}^{U})}{h^{U}}], α \in [h^{L}, h^{U}] \end{matrix}$ $b (α) \in {\begin{matrix} [b_{l} (α), b_{r} (α)] = [a_{4}^{L} - \frac{α (a_{4}^{L} - a_{3}^{L})}{h^{L}}, a_{4}^{U} - \frac{α (a_{4}^{U} - a_{3}^{U})}{h^{U}}], α \in [0, h^{L}] \\ [a_{l} (α), b_{r} (α)] = [a_{1}^{U} + \frac{α (a_{2}^{U} - a_{1}^{U})}{h^{U}}, a_{4}^{U} - \frac{α (a_{4}^{U} - a_{3}^{U})}{h^{U}}], α \in [h^{L}, h^{U}] \end{matrix} .$

Finally for the nearest interval approximation $C (\tilde{A}) = [C_{L}, C_{R}]$ , we have ${\begin{matrix} C_{Ll} = \frac{\int_{0}^{h^{U}} a_{l} (α) d α}{h^{U}} = a_{1}^{U} + \frac{(a_{2}^{U} - a_{1}^{U})}{2} \\ C_{Lr} = \frac{\int_{0}^{h^{L}} a_{r} (α) d α}{h^{L}} = a_{1}^{L} + \frac{(a_{2}^{L} - a_{1}^{L})}{2} \\ C_{Rl} = \frac{\int_{0}^{h^{L}} b_{l} (α) d α}{h^{L}} = a_{4}^{L} - \frac{(a_{4}^{L} - a_{3}^{L})}{2} \\ C_{Rr} = \frac{\int_{0}^{h^{U}} b_{r} (α) d α}{h^{U}} = a_{4}^{U} - \frac{(a_{4}^{U} - a_{3}^{U})}{2} \end{matrix},$

In view of this theorem, we further solve the linear programming model whose parameters are trapezoidal IT2FNs.

3.2 Linear programming model with trapezoidal interval Type-2 fuzzy numbers

In this part, we solve the linear programming model with trapezoidal IT2FNs, which is more close to the complex real situation. Based on the nearest interval approximation of the trapezoidal IT2FNs, the trapezoidal IT2FNs in the model are transformed into interval numbers which both upper and lower limits are interval numbers. Then, with the help of BWC method, the transformed interval linear programming problem is transformed into four sub-models. The solutions of these four models represent four different good or bad cases. The following model describe the solving process.

The basic model of linear programming is as follows: $\begin{matrix} max Z = \sum {\tilde{c}}_{j} x_{j} \\ s . t . \sum {\tilde{a}}_{ij} x_{j} ⩽ {\tilde{b}}_{i}, i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (16)

In this model, $\tilde{a}$ , $\tilde{b}$ and $\tilde{c}$ are trapezoidal IT2FNs.

From theorem 1, it can be known that, for a IT2FN $\tilde{A} = ((a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h^{L}), (a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h^{U}))$ . Its nearest interval approximation is $C_{d} (\tilde{A}) = [C_{L}, C_{R}]$ , where C_L ∈ [C_Ll, C_Lr] , C_U ∈ [C_Rl, C_Rr], $\begin{matrix} C_{Ll} = \frac{\int_{0}^{h^{U}} a_{l} (α) d α}{h^{U}} = a_{1}^{U} + \frac{(a_{2}^{U} - a_{1}^{U})}{2}, \\ C_{Lr} = \frac{\int_{0}^{h^{L}} a_{r} (α) d α}{h^{L}} = a_{1}^{L} + \frac{(a_{2}^{L} - a_{1}^{L})}{2}, \\ C_{Rl} = \frac{\int_{0}^{h^{L}} b_{l} (α) d α}{h^{L}} = a_{4}^{L} - \frac{(a_{4}^{L} - a_{3}^{L})}{2}, \\ C_{Rr} = \frac{\int_{0}^{h^{U}} b_{r} (α) d α}{h^{U}} = a_{4}^{U} - \frac{(a_{4}^{U} - a_{3}^{U})}{2} . \end{matrix},$

By converting all parameters into their nearest interval approximation, the basic model of linear programming can be transformed into: $\begin{matrix} max Z = \sum [\underline{C} \tilde{cj}, \bar{C} \tilde{cj}] x_{j} \\ s . t . \sum [\underline{C} {\tilde{a}}_{ij}, \bar{C} {\tilde{a}}_{ij}] x_{j} ⩽ [\underline{C} {\tilde{b}}_{i}, \bar{C} {\tilde{b}}_{i}], i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (17) where the endpoints of each interval have their own upper and lower limits.

According to the BWC method, the linear programming model can be transformed into two sub-models, representing the best case and the worst case respectively.

1. The best model $\begin{matrix} max Z = \sum \bar{C} {\tilde{c}}_{j} x_{j} \\ s . t . \sum \underline{C} {\tilde{a}}_{ij} x_{j} ⩽ \bar{C} {\tilde{b}}_{i}, i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (18)

2. The worst model $\begin{matrix} max Z = \sum \underline{C} {\tilde{c}}_{j} x_{j} \\ s . t . \sum \bar{C} {\tilde{a}}_{ij} x_{j} ⩽ \underline{C} {\tilde{b}}_{i}, i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (19)

In these two models, each parameter is still an interval, so the two models can be separated into two sub-models respectively. Thus, we can get the following four models:

1. The best-best model $\begin{matrix} max Z_{bb} = \sum (c_{4}^{U} + \frac{(c_{3}^{U} - c_{4}^{U})}{2}) x_{j} \\ s . t . \sum (a_{1}^{U} + \frac{(a_{2}^{U} - a_{1}^{U})}{2}) x_{j} ⩽ (b_{4}^{U} + \frac{(b_{3}^{U} - b_{4}^{U})}{2}), i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (20)

By solving the model we can obtain x_1bb, x_2bb, z_bb.

2. The best-worst model $\begin{matrix} max x_{2 bb} = \sum (c_{4}^{L} + \frac{(c_{3}^{L} - c_{4}^{L})}{2}) x_{j} \\ s . t . \sum (a_{1}^{L} + \frac{(a_{2}^{L} - a_{1}^{L})}{2}) x_{j} ⩽ (b_{4}^{L} + \frac{(b_{3}^{L} - b_{4}^{L})}{2}), i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (21)

By solving the model we can obtain x_1bw, x_2bw, z_bw.

3. The worst-best model $\begin{matrix} max Z_{wb} = \sum (c_{1}^{L} + \frac{(c_{2}^{L} - c_{1}^{L})}{2}) x_{j} \\ s . t . \sum (a_{4}^{L} + \frac{(a_{3}^{L} - a_{4}^{L})}{2}) x_{j} ⩽ (b_{1}^{L} + \frac{(b_{2}^{L} - b_{1}^{L})}{2}), i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (22)

By solving the model we can obtain x_1wb, x_2wb, z_wb.

4. The worst-worst model $\begin{matrix} max Z_{ww} = \sum (c_{1}^{U} + \frac{(c_{2}^{U} - c_{1}^{U})}{2}) x_{j} \\ s . t . \sum (a_{4}^{U} + \frac{(a_{3}^{U} - a_{4}^{U})}{2}) x_{j} ⩽ (b_{1}^{U} + \frac{(b_{2}^{U} - b_{1}^{U})}{2}), i = 1, 2, . . ., m \\ x_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix}$ (23)

By solving the model we can obtain x_1ww, x_2ww, z_ww.

From the above four linear programming problems with crisp values, four solutions of the original model can be obtained and, which describe the best and worst cases of the problem. We can get that Z max = [Z_w, Z_b], where Z_w ∈ [Z_ww, Z_wb] , Z_b ∈ [Z_bw, Z_bb].

Fig. 3

Proposed methodology.

For the results obtained, we make the following understanding.

If the decision maker is more optimistic and more confident, z_bb can be obtained with x_1bw and x_2bb.

If the decision maker is more optimistic, but less confident, z_bw can be obtained with x_1bw and x_2bw.

If the decision maker is more pessimistic, but more confident, z_wb can be obtained with x_1wb and x_2wb.

If the decision maker is more pessimistic and less confident, z_ww can be obtained with x_1ww and x_2ww.

3.3 Discussion

The model proposed in this paper is improved on Javanmard et al. [37]’s method of solving FLP modes in which all parameters are triangle IT2FNs. The differences between the two papers are as follows:

Javanmard et al. [37] mainly solves the FLP modes in which all parameters are triangular IT2FNs. However, there is a certain gap in the study of linear programming models with full trapezoidal IT2FNs which can can represent more uncertainties in real life and are closer to the way people present language. This paper focuses on FLP modes in which all parameters are trapezoidal IT2FNs, which has a wider application scope.

Although the model parameters proposed by Javanmard et al. [37] are triangular IT2FNs, the nearest interval approximation of the parameters is solved from the perspective of type-1 fuzzy numbers. We consider the α-cut of trapezoidal IT2FNs, prove and solve the nearest interval approximation of trapezoidal IT2FNs.

As for the results of solving with the help of BWC method, Javanmard et al. [37] believes that the final solution is formed into a new triangular IT2FNs. However, it is difficult to get this even considering the practical significance and the property of membership function. In this paper, the solution result of the model is still an interval number, which can give some guidance to the decision maker.

3.4 Specific Implementation Steps for Solving a Model Based on the nearest interval approximation

The model building and solving steps mainly include the following 3 steps:

Step 1: Calculate the nearest interval approximation of each parameter to transform FLP models in which all parameters are trapezoidal IT2FNs to interval LP models.

Step 2: Transform interval LP models to 4 sub-models with the BWC method.

Step 3: Solve each sub-model to obtain the final interval number solution.

4 Application examples

A factory will arrange to produce I and II products during the planning period. It is known that the production unit product I needs a₁₁ kg raw materials I and a₂₁ kg raw material II and production unit product II needs a₁₂ kg raw materials I, a₂₂ kg raw material II. The factory now has b₁ kg raw materials I, b₂ kg raw material II. Per unit product I can profit c₁ dollars and per unit product II can profit c₂ dollars. Try to find out how to arrange the production plan to maximize the profit.

Due to the influence of many factors in the actual production process and the instability of the market environment, experts can not give specific raw material consumption, raw material holding and the profitability of the products in the market for each production. Therefore, all parameters are represented by common trapezoidal IT2FNs.

As described in the title, the following linear programming model is listed: $\begin{matrix} max Z = {\tilde{c}}_{1} x_{1} + {\tilde{c}}_{2} x_{2} \\ s . t . {\tilde{a}}_{11} x_{1} + {\tilde{a}}_{12} x_{2} ⩽ {\tilde{b}}_{1} \\ {\tilde{a}}_{21} x_{1} + {\tilde{a}}_{22} x_{2} ⩽ {\tilde{b}}_{2} \\ x_{1}, x_{2} ⩾ 0 \end{matrix}$ (24)

where, $\begin{matrix} {\tilde{a}}_{11} = ((1, \frac{3}{2}, 2, \frac{5}{2}; \frac{2}{3}), (\frac{1}{2}, \frac{5}{4}, \frac{9}{4}, 3; 1)) \\ {\tilde{a}}_{12} = ((\frac{1}{2}, 1, \frac{3}{2}, 2; \frac{2}{3}), (\frac{1}{4}, \frac{3}{4}, \frac{7}{4}, \frac{9}{4}; 1)) \\ {\tilde{a}}_{21} = ((\frac{2}{3}, \frac{4}{3}, 2, \frac{8}{3}; \frac{2}{3}), (\frac{1}{3}, 1, \frac{7}{3}, 3; 1)), \\ {\tilde{a}}_{22} = ((\frac{3}{4}, \frac{5}{4}, \frac{7}{4}, \frac{9}{4}; \frac{2}{3}), (\frac{1}{2}, 1, 2, \frac{5}{2}; 1)), \\ {\tilde{b}}_{1} = ((2, 3, 5, 6; \frac{2}{3}), (\frac{3}{2}, \frac{5}{2}, \frac{11}{2}, \frac{13}{2}; 1)), \\ {\tilde{b}}_{2} = ((2, 4, 6, 8; \frac{2}{3}), (1, 3, 7, 9; 1)), \\ {\tilde{c}}_{1} = ((1, 2, 3, 4; \frac{2}{3}), (\frac{1}{2}, \frac{3}{2}, \frac{7}{2}, \frac{9}{2}; 1)), \\ {\tilde{c}}_{2} = ((\frac{2}{3}, \frac{4}{3}, 2, \frac{8}{3}; \frac{2}{3}), (\frac{1}{3}, 1, \frac{7}{3}, 3; 1)) . \end{matrix}$

The nearest interval approximation of these trapezoidal IT2FNs can be calculated: $\begin{matrix} C {\tilde{a}}_{11} \in [C_{{\tilde{a}}_{11}}^{L}, C_{{\tilde{a}}_{11}}^{R}], where C_{{\tilde{a}}_{11}}^{L} \in [C_{{\tilde{a}}_{11}}^{Ll}, C_{{\tilde{a}}_{11}}^{Lr}] = (\frac{7}{8}, \frac{5}{4}), \\ C_{{\tilde{a}}_{11}}^{R} \in [C_{{\tilde{a}}_{11}}^{Rl}, C_{{\tilde{a}}_{11}}^{Rr}] = (\frac{9}{4}, \frac{21}{8}) \\ C_{{\tilde{a}}_{12}} \in [C_{{\tilde{a}}_{12}}^{L}, C_{{\tilde{a}}_{12}}^{R}], where C_{{\tilde{a}}_{12}}^{L} \in [C_{{\tilde{a}}_{12}}^{Ll}, C_{{\tilde{a}}_{12}}^{Lr}] = (\frac{1}{2}, \frac{3}{4}), \\ C_{{\tilde{a}}_{12}}^{U} \in [C_{{\tilde{a}}_{12}}^{Rl}, C_{{\tilde{a}}_{12}}^{Rr}] = (\frac{7}{4}, 2) \\ C_{{\tilde{a}}_{21}} \in [C_{{\tilde{a}}_{21}}^{L}, C_{{\tilde{a}}_{21}}^{R}], where C_{{\tilde{a}}_{21}}^{L} \in [C_{{\tilde{a}}_{21}}^{Ll}, C_{{\tilde{a}}_{21}}^{Lr}] = (\frac{2}{3}, 1), \\ C_{{\tilde{a}}_{21}}^{U} \in [C_{{\tilde{a}}_{21}}^{Rl}, C_{{\tilde{a}}_{21}}^{Rr}] = (\frac{7}{3}, \frac{8}{3}) \\ C {\tilde{a}}_{22} \in [C_{{\tilde{a}}_{22}}^{L}, C_{{\tilde{a}}_{22}}^{R}], where C_{{\tilde{a}}_{22}}^{L} \in [C_{{\tilde{a}}_{22}^{Ll}, C_{{\tilde{a}}_{22}}^{Lr}}] = (\frac{3}{4}, 1), \\ C_{{\tilde{a}}_{22}}^{R} \in [C_{{\tilde{a}}_{22}}^{Rl}, C_{{\tilde{a}}_{22}}^{Rr}] = (2, \frac{9}{4}) \\ C {\tilde{b}}_{1} \in [C_{\tilde{b} 1}^{L}, C_{\tilde{b} 1}^{R}], where C_{\tilde{b} 1}^{L} \in [C_{{\tilde{b}}_{1}}^{Ll}, C_{\tilde{b} 1}^{Lr}] = (2, \frac{5}{2}), \\ C_{{\tilde{b}}_{1}}^{R} \in [C_{\tilde{b} 1}^{Rl}, C_{\tilde{b} 1}^{Rr}] = (\frac{11}{2}, 6) \\ C {\tilde{b}}_{2} \in [C_{\tilde{b} 2}^{L}, C_{\tilde{b} 2}^{R}], where C_{\tilde{b} 2}^{L} \in [C_{\tilde{b} 2}^{Ll}, C_{\tilde{b} 2}^{Lr}] = (2, 3), \\ C_{{\tilde{b}}_{2}}^{R} \in [C_{\tilde{b} 2}^{Rl}, C_{\tilde{b} 2}^{Rr}] = (7, 8) \\ C {\tilde{c}}_{1} \in [C_{\tilde{c} 1}^{L}, C_{\tilde{c} 1}^{R}], where C_{\tilde{c} 1}^{L} \in [C_{\tilde{c} 1}^{Ll}, C_{\tilde{c} 1}^{Lr}] = (1, \frac{3}{2}), \\ C_{{\tilde{c}}_{1}}^{R} \in [C_{\tilde{c} 1}^{Rl}, C_{\tilde{c} 1}^{Rr}] = (\frac{7}{2}, 4) \end{matrix}$

$\begin{matrix} C {\tilde{c}}_{2} \in [C_{{\tilde{c}}_{1}}^{L}, C_{{\tilde{c}}_{1}}^{R}], where C_{{\tilde{c}}_{1}}^{L} \in [C_{{\tilde{c}}_{1}}^{Ll}, C_{{\tilde{c}}_{1}}^{Lr}] = (\frac{2}{3}, 1), \\ C_{{\tilde{c}}_{1}}^{R} \in [C_{{\tilde{c}}_{1}}^{Rl}, C_{{\tilde{c}}_{1}}^{Rr}] = (\frac{7}{3}, \frac{8}{3}) . \end{matrix}$

According to the BWC method, the following four models can be obtained:

1. The best-best model $\begin{matrix} max Z_{bb} = 4 x_{1} + \frac{8}{3} x_{2} \\ s . t . \frac{7}{8} x_{1} + \frac{1}{2} x_{2} ⩽ 6 \\ \frac{2}{3} x_{1} + \frac{3}{4} x_{2} ⩽ 8 \\ x_{1}, x_{2} ⩾ 0 \end{matrix}$ (25)

By solving the model we can obtain x_1bb = 1.55, x_2bb = 9.29, Z_bb = 30.97.

2. The best-worst model $\begin{matrix} max Z_{bw} = \frac{7}{2} x_{1} + \frac{7}{3} x_{2} \\ s . t . \frac{5}{4} x_{1} + \frac{3}{4} x_{2} ⩽ \frac{11}{2} \\ x_{1} + x_{2} ⩽ 7 \\ x_{1}, x_{2} ⩾ 0 \end{matrix}$ (26)

By solving the model we can obtain x_1bw = 0.5, x_2bw = 6.5, Z_bw = 16.92.

3. The worst-best model $\begin{matrix} max Z_{wb} = \frac{3}{2} x_{1} + x_{2} \\ s . t . \frac{9}{4} x_{1} + \frac{7}{4} x_{2} ⩽ \frac{5}{2} \\ \frac{7}{3} x_{1} + 2 x_{2} ⩽ 3 \\ x_{1}, x_{2} ⩾ 0 \end{matrix}$ (27)

By solving the model we can obtain x_1wb = 1.11, x_2wb = 0, Z_bw = 1.67.

4. The worst-worst model $\begin{matrix} max Z_{ww} = x_{1} + \frac{2}{3} x_{2} \\ s . t . \frac{21}{8} x_{1} + 2 x_{2} ⩽ 2 \\ \frac{8}{3} x_{1} + \frac{9}{4} x_{2} ⩽ 2 \\ x_{1}, x_{2} ⩾ 0 \end{matrix}$ (28)

By solving the model we can obtain x_1ww = 0.75, x_2ww = 0, Z_ww = 0.75.

The above calculation gives the following solution (as shown in Table 1).

Table 1

The results of examples

	The best-best case	The best-worst case	The worst-best case	The worst-worst case
x₁	1.55	0.5	1.11	0.75
x₂	9.29	6.5	0	0
Z	30.97	16.92	1.67	0.75

The results show that in the best case, the production of 1.55 kg products I and 9.29 kg products II can obtain 30.97 dollars profit, while in the worst case, the production of 0.75 kg products I can only profit 0.75 dollars. Moreover, there are two middle cases to weigh and choose.

All parameters are trapezoidal IT2FNs, which can deal with great fuzzy information in real world. If the decision maker is more optimistic and more confident, the factory can arrange to produce 1.55 kg product I and 9.29 kg product II to maximize the profit (30.97 dollars). If the decision maker is more optimistic, the factory can arrange to produce 0.5 kg product I and 6.5 kg product II to maximize the profit (16.92 dollars). If the decision maker is more pessimistic, but more confident, the factory can arrange to produce 1.11 kg product I and 0 kg product II to maximize the profit (1.67 dollars). If the decision maker is more pessimistic and less confident, the factory can arrange to produce 0.75 kg product I and 0 kg product II to maximize the profit (0.75 dollars).

From the whole solution process, the loss of fuzzy information is reduced as much as possible and the model is transformed to interval linear programming model by using the nearest interval approximation. Then, with BWC method, we obtain four sub-models and finally obtain four specific real solutions, which represent the four different kinds of cases and give decision makers a larger reference value.

5 Conclusion

In this study, a method is proposed to solve the linear programming model in which all parameters are trapezoidal interval type-2 fuzzy numbers. In the proposed method, the nearest interval approximation of trapezoidal interval type-2 fuzzy numbers is proved which helps to transform fuzzy linear programming models in which all parameters are trapezoidal interval type-2 fuzzy numbers to interval linear programming models. With the best and worst cases method, Interval linear programming models are transformed to 4 sub-models to obtain four solutions representing different good and bad cases. This method provides a new perspective for the study of type-2 fuzzy linear programming model. The solution process is clear and easy to understand. At present, linear programming model has been widely used in industry, agriculture, business and other fields. This method can be used when parameters involve uncertain information, especially linguistic uncertainty.

At the same time, this method does not consider some limitations of the best and worst cases method, which can not guarantee the solution regions are completely feasible and optimal. Also, the results calculated by this method is not precise enough. Some of the possible future directions are given as follows: Some limitations of the best and worst cases method can be consider carefully. Further the proposed approach can be extended for solving fully fuzzy linear programming problems, where all parameters and variables are trapezoidal interval type-2 fuzzy numbers.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) (71771051, 72071045).

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