Composite Gauss-Legendre Formulas for Solving Fuzzy Integration

Abstract

Two numerical integration rules based on composition of Gauss-Legendre formulas for solving integration of fuzzy numbers-valued functions are investigated in this paper. The methods' constructions are presented and the corresponding convergence theorems are shown in detail. Two numerical examples are given to illustrate the proposed algorithms finally.

1. Introduction

Numerical integration is one of the basic contents in numerical mathematics, and it always plays a vital role in engineering and science calculation. Numerical integration methods are introduced in detail [1]. Numerical integration is always carried out by mechanical quadrature and its basic scheme [2] is as follows: ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈^𝑛_𝑘=0𝐴_𝑘𝑓𝑥_𝑘, (1) where 𝐴_𝑘≥0, 𝑘=0,1,…,𝑛, and 𝑥_𝑘∈[𝑎,𝑏], 𝑘=0,1,…,𝑛, are called coefficients and nodes for mechanical quadrature, respectively. Once the coefficients and nodes are set down, the scheme (1) can be determined.

Over years, some works have emerged about the asymptotic properties of numerical integration methods. However, their results are concise, but reasoning processes are very complicated [3–6]. The topic of fuzzy integration was first discussed in [7]. In 2005, Allahviranloo [8] made a good attempt to use Newton Cot’s methods with positive coefficients for integration of fuzzy functions. For instance, he designed Trapezoidal integration rule and Simpson integration rule for fuzzy integral. Later, they applied the Gaussian quadrature method and Romberg method for approximation of fuzzy integral and fuzzy multiple integral, built a series of formulas for intricate fuzzy integral [8–11], and obtained some good results. But their methods did not have high convergence order.

In this paper, we set up a class of high algebraic accuracy numerical integration methods which are proposed by compositing the two-point and three-point Gauss-Legendre formulas. We design these formulas to calculate integration of fuzzy functions. We also present the methods’ remainder terms and give corresponding convergence theorems. Compared with some approaches for approximating fuzzy integrations before, our methods are superior to those formulas on both amount of calculation and quadrature error. The structure of this paper is as follows.

In Section 2, we recall some basic definitions and results on integration of fuzzy functions. In Section 3, we introduce the two-point and three-point Gauss-Legendre formulas and their composite method. Then we design them to solve fuzzy integration. We also put up methods’ reminder term representations and convergence theorems. The proposed algorithms are illustrated by solving two examples in Section 4 and the conclusion is drawn in Section 5.

2. Preliminaries

2.1. Integration of Fuzzy Function

Let 𝐸¹ be the set of all real fuzzy numbers which are normal, upper semicontinuous, convex, and compactly supported fuzzy sets.

Definition 1 (see [12]). A fuzzy number 𝑢 in parametric form is a pair (𝑢,𝑢) of functions 𝑢(𝑟), 𝑢(𝑟), 0≤𝑟≤1, which satisfies the following requirements:(1)𝑢(𝑟) is a bounded monotonic increasing left continuous function,(2)𝑢(𝑟) is a bounded monotonic decreasing left continuous function,(3)𝑢(𝑟)≤𝑢(𝑟), 0≤𝑟≤1.
Based on Zadeh’s principle of extension, Goetschel et al. [13] presented fuzzy numbers’ addiction and multiplication by 𝑘 which are as follows. Let 𝑥=(𝑥(𝑟),𝑥(𝑟)), 𝑦=(𝑦(𝑟),𝑦(𝑟))∈𝐸¹, 0≤𝑟≤1, and real number 𝑘, as follows:(1)𝑥+𝑦=(𝑥(𝑟)+𝑦(𝑟),𝑥(𝑟)+𝑦(𝑟)),(2)𝑥−𝑦=(𝑥(𝑟)−𝑦(𝑟),𝑥(𝑟)−𝑦(𝑟)),(3)𝑘𝑥=^{(𝑘𝑥(𝑟),𝑘𝑥(𝑟)),𝑘≥0,}_{(𝑘𝑥(𝑟),𝑘𝑥(𝑟)),𝑘<0.}
For arbitrary fuzzy numbers 𝑥=(𝑥(𝑟),𝑥(𝑟)), 𝑦=(𝑦(𝑟),𝑦(𝑟))∈𝐸¹, the quantity 𝐷𝑥,𝑦=sup_0≤𝑟≤1max𝑥(𝑟)−𝑦(𝑟),𝑥(𝑟)−𝑦(𝑟) (2) is the distance between 𝑥 and 𝑦. A function 𝑓∶𝑅¹→𝐸¹ is called a fuzzy function. If for arbitrary fixed 𝑡₀∈𝑅¹ and 𝜖>0, a 𝛿>0 such that 𝑡−𝑡₀<𝛿⟹𝐷𝑓(𝑡),𝑓𝑡₀<𝜖, (3) exists, 𝑓 is said to be continuous.

Definition 2 (see [14]). Assume 𝑓∶[𝑎,𝑏]→𝐸¹. For each partition 𝑃={𝑡₀,𝑡₁,…,𝑡_𝑛} of [𝑎,𝑏] and for arbitrary 𝜉_𝑖∶𝑡_𝑖−1≤𝜉_𝑖≤𝑡_𝑖, 1≤𝑖≤𝑛, let 𝑅_𝑝=^𝑛_𝑖=1𝑓𝜉_𝑖𝑡_𝑖−𝑡_𝑖−1. (4)
The definite integral of 𝑓(𝑡) over [𝑎,𝑏] is ^𝑏_𝑎𝑓(𝑡)𝑑𝑡=lim𝑅_𝑝,max_{1≤𝑖≤𝑛}𝑡_𝑖−𝑡_𝑖−1⟶0, (5) provided that this limit exists in the metric 𝐷. If the fuzzy function 𝑓(𝑡) is continuous in the metric 𝐷, its definite integral exists. Furthermore, ^𝑏_𝑎𝑓(𝑡;𝑟)𝑑𝑡=^𝑏_𝑎𝑓(𝑡;𝑟)𝑑𝑡,_𝑏𝑎𝑓(𝑡;𝑟)𝑑𝑡=^𝑏_𝑎𝑓(𝑡;𝑟)𝑑𝑡. (6)
It should be noted that the fuzzy integral also can be defined using the Lebesgue-type approach [15, 16]. More details about properties of fuzzy integral are given in [17].

2.2. Gauss-Legendre Formulas

Gauss quadrature formula is the highest algebraic accuracy of interpolation quadrature formula. By reasonably selecting quadrature nodes and quadrature coefficients of the form of ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈^𝑛_𝑘=0𝐴_𝑘𝑓𝑥_𝑘, (7) we can obtain the interpolation quadrature formula with the highest algebraic accuracy; that is, 2𝑛+1. Using root nodes of 𝑛+1 order Legendre orthogonal polynomial on special interval [−1,1], we can propose Gauss-Legendre quadrature formula (7).

Lemma 3 (see [6]). The reminder term of Gauss-Legendre quadrature formula (7) is 𝐸𝑓=^𝑏_𝑎𝑓(𝑥)𝑑𝑥−^𝑛_𝑘=0𝐴_𝑘𝑓𝑥_𝑘=^𝑏_𝑎𝑓^(2𝑛+2)(𝜉)()!𝜔2𝑛+2²(𝑥)𝑑𝑥,𝜉∈[𝑎,𝑏], (8) where 𝜔(𝑥)=∏^𝑛_𝑖=0(𝑥−𝑥_𝑖) and 𝑥_𝑖 are Gauss nodes.
In particular, when 𝑛=1, (7) is two-point Gauss-Legendre quadrature formula: ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈𝑏−𝑎2𝑓𝑥−√33𝑥√+𝑓33, (9) where 𝑥(𝑡)=(𝑎+𝑏)/2+((𝑏−𝑎)/2)𝑡 and its reminder term is 𝐸𝑓=^𝑏_𝑎𝑓⁽⁴⁾(𝜉)−√4!𝑥−𝑥33²√𝑥−𝑥33²[].𝑑𝑥,𝜉∈𝑎,𝑏 (10)
When 𝑛=2, (7) is three-point Gauss-Legendre quadrature formula: ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈𝑏−𝑎259𝑓𝑥−√155+89𝑓𝑎+𝑏2+59𝑓𝑥√155, (11) and its reminder term is 𝐸𝑓=^𝑏_𝑎𝑓⁽⁶⁾(𝜉)−√6!𝑥−𝑥155²×𝑥−𝑎+𝑏2²√𝑥−𝑥155²[].𝑑𝑥,𝜉∈𝑎,𝑏 (12)

2.3. Convergence Order of Composite Method

Definition 4 (see [18]). Suppose 𝐼=∫^𝑏_𝑎𝑓(𝑥)𝑑𝑥, and 𝐼_𝑛 is a composite numerical integration method. If ℎ→0, it satisfies lim_ℎ→0𝐼−𝐼_𝑛ℎ^𝑝=𝐶,𝐶≠0, (13) and we call 𝐼_𝑛 a 𝑝 order convergent method.
For instance, composite Trapezoid method ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈𝑏−𝑎2^𝑛−1_𝑘=0𝑓𝑥_𝑖+𝑓𝑥_𝑖+1 (14) and composite Simpson ^𝑏_𝑎𝑓(𝑥)ℎ𝑑𝑥≈6^𝑛−1_𝑘=0𝑓𝑥_𝑖+4𝑓𝑥_𝑖+1/2+𝑓𝑥_𝑖+1 (15) have two-order and four-order convergence property, respectively.

3. Gauss-Legendre Formulas for Solving Fuzzy Integral

3.1. Composite Formulas and Their Error Reminders

Theorem 5. Let 𝑓(𝑥)∈𝐶⁽⁶⁾[𝑎,𝑏], 𝑥_𝑘=𝑥₀+𝑘ℎ, 𝑘=0,1,2,…,𝑛, ℎ=(𝑏−𝑎)/𝑛; then the reminder term of composite three-point formulas Gauss-Legendre ^𝑏_𝑎𝑓(𝑥)ℎ𝑑𝑥≈2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+8109𝑓𝑥_𝑖+1/2+59𝑓𝑥_𝑖+1/2+√15ℎ10 (16) is 𝐸𝑓(𝑥)=(𝑏−𝑎)ℎ⁶2⁶𝑓×31500⁽⁶⁾(𝜉),𝜉∈[𝑎,𝑏], (17) and it has six-order convergence property.

Proof. We first consider the remainder term of three-point Gauss-Legendre. By Lemma 3, 𝐸𝑔=¹₋₁𝑔(𝑡)5𝑑𝑡−9𝑔−√155+89𝑔(0)+59𝑔√155=𝑐𝑔⁽⁶⁾(𝜉),𝜉∈[−1,1]. (18) Let 𝑔(𝑡)=𝑡⁶, and 𝑔⁽⁶⁾(𝑡)≡720,¹₋₁𝑡⁶2𝑑𝑡=7,𝐼𝑡⁶=59−√155⁶+59−√155⁶=6.25 (19) We have 𝐸𝑡⁶=¹₋₁𝑡⁶𝑑𝑡−𝐼𝑡⁶=27−625=720𝑐, (20) and 𝑐=1/15750. Thus, we obtain the reminder term of three-point formulas Gauss-Legendre: 𝐸𝑔=1𝑔15750⁽⁶⁾(𝜉),𝜉∈[−1,1]. (21)
Now we study the integral remainder term of composite three-point Gauss-Legendre: ^𝑏_𝑎𝑓(𝑥)𝑑𝑥=^𝑛−1_𝑖=0^𝑥𝑖+1_𝑥𝑖𝑓(𝑥)𝑑𝑥, (22) where 𝑃={𝑥₀,𝑥₁,…,𝑥_𝑛} is about equidistant partition on [𝑎,𝑏]. Let 𝑥(𝑡)=(𝑥_𝑖+𝑥_𝑖+1)/2+(ℎ/2)𝑡, 𝑥_𝑖+1/2=(𝑥_𝑖+𝑥_𝑖+1)/2; then ^𝑥𝑖+1_𝑥𝑖𝑓(𝑥)ℎ𝑑𝑥=2¹₋₁𝑓𝑥_𝑖+1/2+ℎ2𝑡𝑑𝑡. (23) By means of three-point Gauss-Legendre formula and its reminder term (21), we have ^𝑏_𝑎𝑓(𝑥)𝑑𝑥=^𝑛−1_𝑖=0ℎ2¹₋₁𝑓𝑥_𝑖+1/2+ℎ2𝑡=ℎ𝑑𝑡2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√3ℎ2√5+89𝑓𝑥_𝑖+1/2+59𝑓𝑥_𝑖+1/2+√3ℎ2√5+ℎ⁶2⁶𝑓×15750⁽⁶⁾𝜉_𝑖,𝜉_𝑖∈𝑥_𝑖,𝑥_𝑖+1,=ℎ2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√3ℎ2√5+89𝑓𝑥_𝑖+1/2+59𝑓𝑥_𝑖+1/2+√3ℎ2√5+ℎ⁷2^7𝑛−1_𝑖=01𝑓15750⁽⁶⁾𝜉_𝑖,=ℎ2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√3ℎ2√5+89𝑓𝑥_𝑖+1/2+59𝑓𝑥_𝑖+1/2+√3ℎ2√5+(𝑏−𝑎)ℎ⁶2⁶𝑓×31500⁽⁶⁾(𝜉),𝜉∈[𝑎,𝑏]. (24) So we get the reminder term of composite three-point Gauss-Legendre as follows: 𝐸𝑓(𝑥)=(𝑏−𝑎)ℎ⁶2⁶𝑓×31500⁽⁶⁾(𝜉),𝜉∈[𝑎,𝑏]. (25)
Since lim_ℎ→0𝐼−𝐼_𝑛ℎ⁶=(𝑏−𝑎)2⁶𝑓×31500⁽⁶⁾(𝜉), (26) we know it has six-order convergence property.

In similar way, we obtain the reminder term of composite two-point formulas Gauss-Legendre as follows.

Theorem 6. Let 𝑓(𝑥)∈𝐶⁽⁴⁾[𝑎,𝑏], 𝑥_𝑘=𝑥₀+𝑘ℎ, 𝑘=0,1,2,…,𝑛, ℎ=(𝑏−𝑎)/𝑛; then the reminder term of composite two-point formulas Gauss-Legendre ^𝑏_𝑎𝑓(𝑥)≈ℎ𝑑𝑥2^𝑛−1_𝑘=0𝑓𝑥_𝑖+1/2−√3ℎ6+𝑓𝑥_𝑖+1/2+√3ℎ6, (27) is 𝐸𝑓(𝑥)=(𝑏−𝑎)ℎ⁴2⁴𝑓×270⁽⁴⁾(𝜉),𝜉∈[𝑎,𝑏], (28) and it has four-order convergence property.

3.2. Gauss-Legendre Formulas to Fuzzy Integration

In this subsection, we apply composite Gauss-Legendre formulas to solve fuzzy integration and give their reminder terms and convergence theorems.

Applying formulas (16) and (27) to numerical integration for fuzzy function (13), we have ^𝑏_𝑎𝑓()ℎ𝑥;𝑟𝑑𝑥≈2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+810;𝑟9𝑓𝑥_𝑖+1/2+5;𝑟9𝑓𝑥_𝑖+1/2+√15ℎ,10;𝑟_𝑏𝑎𝑓()ℎ𝑥;𝑟𝑑𝑥≈2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+810;𝑟9𝑓𝑥_𝑖+1/2+5;𝑟9𝑓𝑥_𝑖+1/2+√15ℎ,10;𝑟0≤𝑟≤1_𝑏𝑎𝑓()ℎ𝑥;𝑟𝑑𝑥≈2^𝑛−1_𝑘=0𝑓𝑥_𝑖+1/2−√3ℎ6;𝑟+𝑓𝑥_𝑖+1/2+√3ℎ6;𝑟_𝑏𝑎,𝑓()ℎ𝑥;𝑟𝑑𝑥≈2^𝑛−1_𝑘=0𝑓𝑥_𝑖+1/2−√3ℎ6+;𝑟𝑓𝑥_𝑖+1/2+√3ℎ6;𝑟,0≤𝑟≤1. (29)(30)

Theorem 7. Suppose 𝑓(𝑥;𝑟)∈𝐶⁶[𝑎,𝑏] about 𝑥 and 𝑓⁽⁶⁾(𝑥;𝑟)≥0, 𝑓⁽⁶⁾(𝑥;𝑟)≥0, 0≤𝑟≤1; then the reminder terms of composite three-point Gauss-Legendre formulas (29) for fuzzy integration are 𝐸𝑓()=()ℎ𝑥;𝑟𝑏−𝑎⁶2⁶𝑓×31500⁽⁶⁾𝜉,𝐸;𝑟𝑓()=()ℎ𝑥∶𝑟𝑏−𝑎⁶2⁶×31500𝑓⁽⁶⁾𝜉∶𝑟, (31) where 𝜉,𝜉∈[𝑎,𝑏], 0≤𝑟≤1.

Proof. From formulas (16) and (17), we get ^𝑏_𝑎𝑓(𝑥)ℎ𝑑𝑥=2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+8109𝑓𝑥_𝑖+1/2+59𝑓𝑥_𝑖+1/2+√15ℎ+()ℎ10𝑏−𝑎⁶2⁶𝑓×31500⁽⁶⁾(𝜉),𝜉∈[𝑎,𝑏]. (32)
Using the above formula for fuzzy integration in parametric form, we have ^𝑏_𝑎𝑓()ℎ𝑥;𝑟𝑑𝑥=2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+810;𝑟9𝑓𝑥_𝑖+1/2+5;𝑟9𝑓𝑥_𝑖+1/2+√15ℎ+()ℎ10;𝑟𝑏−𝑎⁶2⁶𝑓×31500⁽⁶⁾𝜉;𝑟_𝑏𝑎,𝑓()ℎ𝑥;𝑟𝑑𝑥=2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+810;𝑟9𝑓𝑥_𝑖+1/2+5;𝑟9𝑓𝑥_𝑖+1/2+√15ℎ+()ℎ10;𝑟𝑏−𝑎⁶2⁶×31500𝑓⁽⁶⁾𝜉∶𝑟, (33) where 𝜉,𝜉∈[𝑎,𝑏], 0≤𝑟≤1.
So the reminder terms of composite three-point Gauss-Legendre formulas (29) for fuzzy integration are (31).

Theorem 8. Let 𝑓(𝑥;𝑟)∈𝐶⁶[𝑎,𝑏] about 𝑥, 0≤𝑟≤1; then lim_ℎ→0ℎ2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+810;𝑟9𝑓𝑥_𝑖+1/2+5;𝑟9𝑓𝑥_𝑖+1/2+√15ℎ=10;𝑟_𝑏𝑎𝑓(𝑥;𝑟)𝑑𝑥,lim_ℎ→0ℎ2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+810;𝑟9𝑓𝑥_𝑖+1/2+5;𝑟9𝑓𝑥_𝑖+1/2+√15ℎ=10;𝑟_𝑏𝑎𝑓(𝑥;𝑟)𝑑𝑥. (34)

Proof. From Theorem 7, we have 𝐸𝑓(𝑥;𝑟)=^𝑏_𝑎𝑓()−ℎ𝑥;𝑟𝑑𝑥2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+810;𝑟9𝑓𝑥_𝑖+1/2+5;𝑟9𝑓𝑥_𝑖+1/2+√15ℎ=()ℎ10;𝑟𝑏−𝑎⁶2⁶𝑓×31500⁽⁶⁾𝜉,𝐸;𝑟𝑓(𝑥∶𝑟)=^𝑏_𝑎𝑓()−ℎ𝑥;𝑟𝑑𝑥2^𝑛−1_𝑖=059𝑓𝑥_𝑖+1/2−√15ℎ+810;𝑟9𝑓𝑥_𝑖+1/2+5;𝑟9𝑓𝑥_𝑖+1/2+√15ℎ=()ℎ10;𝑟𝑏−𝑎⁶2⁶×31500𝑓⁽⁶⁾𝜉∶𝑟, (35) where 𝜉,𝜉∈[𝑎,𝑏], 0≤𝑟≤1.
Since 𝑓⁽⁶⁾(𝑥;𝑟), 𝑓⁽⁶⁾(𝑥;𝑟) are bounded over [𝑎,𝑏], we easily get the following fact: 𝐸𝑓;𝑟⟶0,𝐸𝑓;𝑟⟶0, (36) if ℎ→0.

Similarly, we have the following convergence theorem.

Theorem 9. Suppose 𝑓(𝑥;𝑟)∈𝐶⁴[𝑎,𝑏] about 𝑥 and 𝑓⁽⁴⁾(𝑥;𝑟)≥0, 𝑓⁽⁴⁾(𝑥;𝑟)≥0, 0≤𝑟≤1; then the reminder terms of composite two-point Gauss-Legendre formulas (30) for fuzzy integration are 𝐸𝑓()=()ℎ𝑥;𝑟𝑏−𝑎⁴2⁴𝑓×270⁽⁴⁾𝜉,𝐸;𝑟𝑓()=()ℎ𝑥∶𝑟𝑏−𝑎⁴2⁴×270𝑓⁽⁴⁾𝜉∶𝑟, (37) where 𝜉,𝜉∈[𝑎,𝑏], 0≤𝑟≤1. And lim_ℎ→0ℎ2^𝑛−1_𝑘=0𝑓𝑥_𝑖+1/2−√3ℎ6;𝑟+𝑓𝑥_𝑖+1/2+√3ℎ6=;𝑟_𝑏𝑎𝑓(𝑥;𝑟)𝑑𝑥,lim_ℎ→0ℎ2^𝑛−1_𝑘=0𝑓𝑥_𝑖+1/2−√3ℎ6;𝑟+𝑓𝑥_𝑖+1/2+√3ℎ6;𝑟=^𝑏_𝑎𝑓(𝑥;𝑟)𝑑𝑥. (38)

4. Numerical Examples

Example 1. Consider the following fuzzy integral: ²₀𝑘𝑥⁴𝑑𝑥,𝑘=(𝑟,2−𝑟). (39) The exact solution is (32/5)(𝑟,2−𝑟)=6.400000(𝑟,2−𝑟).
From the two-point Gauss-Legendre formula: 𝑄𝑓;𝑟=6.222222𝑟,𝑄(),𝐸𝑓𝑓;𝑟=6.2222222−𝑟;𝑟=0.177778𝑟,𝐸𝑓;𝑟=0.177778(2−𝑟), (40) it is clear that formula (37) holds.
By the two-point Gauss-Legendre formula with ℎ=1𝑄𝑓;𝑟=6.388888𝑟,𝑄(),𝐸𝑓𝑓;𝑟=6.3888882−𝑟;𝑟=0.011112𝑟,𝐸𝑓;𝑟=0.011112(2−𝑟), (41) it is clear that formula (37) holds. Now with ℎ=1/2, 𝑄𝑓;𝑟=6.399305𝑟,𝑄(),𝐸𝑓𝑓;𝑟=6.3993052−𝑟;𝑟=0.000695𝑟,𝐸𝑓;𝑟=0.000695(2−𝑟). (42) Equation (38) holds too.

Example 2. Consider the following fuzzy integral: ¹₀𝑘𝑒^−𝑥𝑑𝑥,𝑘=(−1+𝑟,1−𝑟). (43) The exact solution is (1−1/𝑒)(−1+𝑟,1−𝑟)=0.63212055⋯(−1+𝑟,1−𝑟).
We calculate numerically the above integral using Trapezoidal formula, Simpson formula, composite two-point Gauss-Legendre, and three-point Gauss-Legendre methods with ℎ=1, ℎ=1/2, and ℎ=1/4. Some comparisons about the numerical solutions and the errors between the different methods are shown in Tables 1, 2, and 3. All data are denoted with eight-bit significant digits and errors are calculated by the distance between exact solution and numerical solution.
From the above tables' figures, we can clearly see that our methods have better approximation than the Trapezoidal formula and Simpson formula on the same fuzzy integration, in which the composite three-point Gauss-Legendre is really the case.

Table 1 

Numerical solutions and errors between different methods (ℎ=1).

Table 2 

Numerical solutions and errors between different methods (ℎ=0.5).

Table 3 

Numerical solutions and errors between different methods (ℎ=0.25).

5. Conclusion

In this work, we applied composite Gauss-Legendre formulas to solve fuzzy integral over a finite interval [𝑎,𝑏]. Since this integration yields fuzzy number in parametric form, we use the parametric form of the methods. The integration of triangular fuzzy number is a triangular fuzzy number. Numerical examples showed that our methods are practical and efficient while computing fuzzy integral on a larger interval [𝑎,𝑏].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work is supported by the Natural Scientific Funds of China (nos. 61262022 and 21175108) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-1120).