International Journal of Computational Intelligence Systems

Volume 14, Issue 1, 2021, Pages 1541 - 1563

The Arithmetic Operator of Fuzzy Regular Prismoid Numbers and Its Application to Fuzzy Risk Analysis

Authors
Shexiang Hai*
School of Science, Lanzhou University of Technology, Lanzhou, 730050, P.R. China
*Corresponding author. Email: haishexiang@lut.cn
Corresponding Author
Shexiang Hai
Received 30 December 2020, Accepted 10 April 2021, Available Online 6 May 2021.
DOI
10.2991/ijcis.d.210422.001How to use a DOI?
Keywords
Regular prismoid numbers; The arithmetic operator; Fuzzy regular prismoid numbers approximation; The degree of similarity; Fuzzy risk analysis
Abstract

We devote to study the arithmetic operator of fuzzy regular prismoid numbers as well as the degree of similarity between fuzzy regular prismoid numbers, and then the arithmetic operator and the degree of similarity are applied in risk analysis. Firstly, the arithmetic operator of fuzzy regular prismoid numbers are researched, and some properties of the arithmetic operators are discussed. At the same time, the fuzzy regular prismoid numbers approximation of 2-dimensional fuzzy numbers is discussed as an indispensable part to study the fuzzy risk analysis. Then, the degree of similarity between fuzzy regular prismoid numbers are deeply researched, which are the basic work of studying the fuzzy risk analysis. Finally, the proposed results are applied in risk analysis of a company.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Fuzzy numbers are a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. Since Zadeh [1] put forward the concept of fuzzy numbers to describe the properties of probability functions in 1975, many researchers have been involved in the development of various aspects of the theory and applications of fuzzy numbers. With further study of the theory and applications of fuzzy numbers, the following forms of fuzzy numbers are produced respectively. To study the operations of fuzzy numbers, Dubois and Prade [2] proposed the conception of L-R fuzzy number in 1978. L-R fuzzy numbers as the most general form of fuzzy numbers have been used extensively [3]. With the extensive use of fuzzy numbers, Kaufmann and Gupta [4] introduced the conception of triangular fuzzy number to solve the economic problems. In order to present a comparison method for fuzzy numbers Cheng [5] presented the notion of trapezoid fuzzy number in 1998. As a natural generalization of L-R fuzzy numbers from one-dimension to n-dimension, Wang and Wu [6] generated the concept of fuzzy n-cell numbers in 2002. Later, Wang et al. [7] extended to a fuzzy n-ellipsoid numbers to solve the problem of uncertainty multichannel digital information. For reasons of simplifying the calculation of 2-dimensional fuzzy numbers, Hai et al. [8] put forward the concept of fuzzy 2-cell prismoid numbers in 2020. With the wide application of fuzzy numbers, a large number of outstanding results are constantly emerging. For instance, in view of the fuzzy sets and fuzzy numbers may have some degree of uncertainty and error when available data either come from unreliable sources or refer to events in the future, Seiti et al. [9] presented a novel concept R-numbers for a better modeling of the risks and errors associated with fuzzy numbers. In R-numbers, the membership function has not been taken into account in risk modeling of the fuzzy sets, then Seiti et al. [10] suggested a concept called R-sets in 2021.

The arithmetic operators as the root in the theory and application for fuzzy numbers were discussed by enormous researchers [11,12]. In 1975, Zadeh [13] first proposed the arithmetic operations on fuzzy numbers by the use of the extension principle. Dubois and Prade [14] made used of discretized fuzzy numbers to successfully give the arithmetical operations of fuzzy numbers which is the exact analytical fuzzy operations. Later, Dubois and Prade [15] discussed the nonlinear operations of fuzzy multiplication and fuzzy division on triangular fuzzy numbers by the interval arithmetic. The arithmetic operations of fuzzy numbers are usually approached either by use of the interval arithmetic or the extension principle. The interval fuzzy arithmetic is the most common approach in different applications, due to its simplicity and the availability of computational methods [16]. Based on Archimedean t-norms, Wagenknech et al. [17] derived some formulas for the inclusion of LR-number arithmetic operations. In 2005, Guerra and Stefanin [18] studied the arithmetic operations on fuzzy numbers by α-cut. In the category-theoretic point of view Bica [19] obtained a construction of additive and multiplicative structures for fuzzy numbers in 2007. Considering that the Hukuhara difference [20] is not invertible operations and appears to have several limitations and restrictive, Stefanini [21] proposed the difference and division of fuzzy numbers as inverse operations to addition and multiplication. In 2014, Cano et al. [22] proposed a single- level constrained fuzzy arithmetic which is interval arithmetic on α-levels. Based on the summation and multiplication for random variables, Stupnanova introduced the arithmetics of fuzzy numbers in [23]. The interval fuzzy arithmetic is criticized for accumulation of fuzziness, a phenomenon that causes overestimation of uncertainties in the resulting [24]. This problem can be reduced by the extension principle fuzzy arithmetic [25]. Existing computational methods extension principle fuzzy arithmetic are mainly divided into three methods: min t-norm, drastic product t-norm as well as product and Lukasiewicz t-norms. Klir [26] introduced an extended fuzzy arithmetic using the min t-norm on trapezoidal fuzzy numbers. Mesiar [27] as well as Hong and Do [28] developed a computational method for implementing extended fuzzy arithmetic using the drastic product t-norm on triangular fuzzy numbers. In order to reduce the uncertainty overestimation problem, Seresht and Fayek [25] introduced an extended fuzzy arithmetic using product and Lukasiewicz t-norms on triangular fuzzy numbers. In view of the arithmetical operations based on Zadeh's extension principle may no longer be preserve the basic operational properties of fuzzy numbers, Holcapek and Stepnicka [29] presented a novel framework for arithmetics of extensional fuzzy numbers that preserved the important algebraic properties of the arithmetic of real numbers.

As part of applications for fuzzy numbers, researchers began to study fuzzy risk analysis [30]. With the development of theory and applications of fuzzy risk analysis, there are more and more researchers have devoted to study it [3134]. Just as important, the similarity measure between fuzzy numbers was studied by many researchers to measure similarity [3539]. All of these researches are concentrated on 1-dimensional fuzzy numbers. In many instances, it becomes more reasonable that the practical problem is described by n-dimensional fuzzy numbers [40,41]. For instance, success of a company results in part from its working efficiency of employee which based on their production speed and the quality of their products. If the production speed and the quality of products are denoted by x and y, respectively, then the working efficiency of the employee can be characterized by two-dimension quantity (x,y). If the quantity is an estimated quantity, then using a two-dimension fuzzy number to express the business management is more suitable than a crisp two-dimension quantity. For example, if the production speed and the quality of products are about 0.85 and 0.985 respectively, then the person's working state can be expressed by the 2-dimension fuzzy number as following:

ũ(x,y)={x0.720.06,0.72x0.78,0.05x+0.01980.06y0.01x+0.06720.06,y0.930.05,0.93y0.98,0.06y0.01980.05x0.05y+0.0950.05,1,0.78<x<0.92,0.98<y<0.99,0.97x0.05,0.92x0.97,0.05x0.0950.05y0.01x+0.04030.05,1y0.01,0.99y1,0.06y0.060.01x0.05y+0.04030.01,0,otherwise.

Then the working efficiency of employee can be expressed by the 2-dimensional fuzzy number which is named fuzzy regular prismoid number due to its spatial graph in this paper.

The main tasks of dealing with fuzzy risk analysis are two aspects. One is arithmetic operators of fuzzy numbers. Usually the arithmetic operator of fuzzy regular prismoid numbers is not a fuzzy regular prismoid numbers. How to solve this problem, it becomes particularly important and urgent. The other aspect is the similarity of the fuzzy numbers. The purpose of this paper is to study the arithmetic operators and the similarity of fuzzy regular prismoid numbers, which are used to handle fuzzy risk analysis problems. From this perspective, the presented results should be useful for fuzzy risk analysis. The paper is organized as following. Firstly, arithmetic operators of fuzzy regular prismoid numbers are defined, which are the basic work of studying fuzzy risk analysis. Meanwhile, the fuzzy regular prismoid numbers approximation of 2-dimensional fuzzy numbers is discussed. In Section 3, we start by considering the radius of gyration for fuzzy regular prismoid numbers, and then based on distance and the radius of gyration for fuzzy regular prismoid numbers a similarity measure for fuzzy regular prismoid numbers is being proposed. Finally, we use the regular prismoid numbers to deal with fuzzy risk analysis problems in Section 4.

2. THE ARITHMETIC OPERATION AND FUZZY REGULAR PRISMOID NUMBERS APPROXIMATION OF 2-DIMENSIONAL FUZZY NUMBERS

Throughout this study, F(Rn) denotes the set of all fuzzy subsets on the n-dimensional Euclidean space Rn. If ũF(Rn),r(0,1], then we write the r-level sets of ũ as [ũ]r={xRn:ũ(x)r}. Suppose that ũF(Rn), satisfies the following conditions:

  1. ũ is a normal fuzzy set, i.e., an x0Rn exists such that ũ(x0)=1,

  2. ũ is a convex fuzzy set, i.e., ũ(λx+(1λ)y)min{ũ(x),ũ(y)} for any x,yRn and λ[0,1],

  3. ũ is upper semicontinuous,

  4. [ũ]0={xRn:ũ(x)>0}¯=r(0,1][ũ]r¯ is compact, where A¯ denotes the closure of A.

Then ũ is called a fuzzy number. We use En to denote the fuzzy number space [42].

It is clear that each uRn can be considered as a fuzzy number ũ defined by

u~(x)={1,x=u0,otherwise.

Suppose that ũ,ṽE2, and the membership function of the projective for ũ,ṽ on the xoz and yoz plane satisfy

p̃u(x)={Lu(x),axx<bx,1,bxxcx,Ru(x),cx<xdx,0,otherwise,
p̃u(y)={Lu(y),ayy<by,1,byycy,Ru(y),cy<ydy,0,otherwise,
p̃v(x)={Lv(x),axx<bx,1,bxxcx,Rv(x),cx<xdx,0,otherwise,
p̃v(y)={Lv(y),ayy<by,1,byycy,Rv(y),cy<ydy,0,otherwise.

Let Rp̃u(x)1(r),Rp̃v(x)1(r),Lp̃u(x)1(r),Lp̃v(x)1(r),Rp̃u(y)1(r),Rp̃v(y)1(r),Lp̃u(y)1(r) and Lp̃v(y)1(r) be the inverse function of Ru(x),Rv(x),Lu(x),Lv(x),Ru(y),Rv(y),Lu(y) and Lv(y), respectively. The weighted pseudometric D:E2×E2[0,+) between ũ and ṽ is defined by

D(u~,υ~)=1201f(r)(Rp̃u(x)1(r)Rp̃v(x)1(r))2dr+01f(r)(Rp̃u(y)1(r)Rp̃v(y)1(r))2dr12+01f(r)(Lp̃u(x)1(r)Lp̃v(x)1(r))2dr+01f(r)(Lp̃u(y)1(r)Lp̃v(y)1(r))2dr12,
where the function f(r) is increasing on [0,1], f(r)>0 for all r(0,1] with f(0)=0 and 01f(r)dr=12.

The trapezoidal fuzzy number as a special fuzzy number is widely used in fuzzy nonlinear regression, fuzzy decision-making, fuzzy optimization and other fields [4345]. However, it becomes more reasonable that the practical problem is described by n-dimensional fuzzy numbers. We generalize the trapezoidal fuzzy number to 2-dimension and propose the concept of fuzzy regular prismoid number as follows.

Definition 1.

Let ũE2,a1,a2,a3,a4,b1,b2,b3,b4R,a1a2a3a4 and b1b2b3b4. If the membership function of ũ can be defined as

ũ(x,y)={xa1a2a1,a1xa2,(b2b1)x+a2b1a1b2a2a1y(b3b4)x+a2b4a1b3a2a1,yb1b2b1,b1yb2,(a2a1)y+a1b2a2b1b2b1x(a3a4)y+a4b2a3b1b2b1,1,a2xa3,b2yb3,a4xa4a3,a3xa4,(b2b1)x+a3b1a4b2a3a4y(b3b4)x+a3b4a4b3a3a4,b4yb4b3,b3yb4,(a2a1)y+a1b3a2b4b3b4x(a3a4)y+a4b3a3b4b3b4,0,otherwise.

Then ũ is called a fuzzy regular prismoid number. For brevity, the fuzzy regular prismoid number are denoted as ũ=(a1,a2,a3,a4;b1,b2,b3,b4). We use P(E2) to denote the fuzzy regular prismoid number space.

Let ũP(E2). For any r(0,1],

[ũ]r=(a2a1)r+a1,a4(a4a3)r×(b2b1)r+b1,b4(b4b3)r.

Given ũ=(a1,a2,a3,a4;b1,b2,b3,b4) and ṽ=(c1,c2,c3,c4;d1,d2,d3,d4) are both fuzzy regular prismoid numbers, the weighted pseudometric D:P(E2)×P(E2)[0,+) between ũ and ṽ is

D(u~,υ~)=1201f(r)(Rp̃u(x)1(r)Rp̃v(x)1(r))2dr+01f(r)(Rp̃u(y)1(r)Rp̃v(y)1(r))2dr12+01f(r)(Lp̃u(x)1(r)Lp̃v(x)1(r))2dr+01f(r)(Lp̃u(y)1(r)Lp̃v(y)1(r))2dr12=1201f(r)(a4c4(a4c4a3+c3)r)2dr+01f(r)(b4d4(b4d4b3+d3)r)2dr12+01f(r)((a2a1c2+c1)r+a1c1)2dr+01f(r)((b2b1d2+d1)r+b1d1)2dr12.

2.1. The Arithmetic Operation of 2-Dimensional Fuzzy Numbers

The arithmetic operation play an important role in fuzzy risk analysis, hereafter the arithmetic operation for the 2-dimensional fuzzy numbers will be researched.

Definition 2.

Let ũ,ṽE2. If [ũ]r=[u1(r),u1+(r)]×[u2(r),u2+(r)] and [ṽ]r=[v1(r),v1+(r)]×[v2(r),v2+(r)] for any r[0,1], then the arithmetic operation for ũ and ṽ are defined as follows:

  1. Addition of the fuzzy regular prismoid numbers satisfies

    [u~υ~]r=infβrminλ[0,1]{(λu1(β)+(1λ)u1+(β))+(λv1(β)+(1λ)v1+(β))},supβrmaxλ[0,1]{(λu1(β)+(1λ)u1+(β))+(λv1(β)+(1λ)v1+(β))}×infβrminλ[0,1]{(λu2(β)+(1λ)u2+(β))+(λv2(β)+(1λ)v2+(β))},supβrmaxλ[0,1]{(λu2(β)+(1λ)u2+(β))+(λv2(β)+(1λ)v2+(β))},
    for any r[0,1].

  2. Subtraction of the fuzzy regular prismoid numbers satisfies

    [u~υ~]r=infβrminλ[0,1]{(λu1(β)+(1λ)u1+(β))(λv1(β)+(1λ)v1+(β))},supβrmaxλ[0,1]{(λu1(β)+(1λ)u1+(β))(λv1(β)+(1λ)v1+(β))}×infβrminλ[0,1]{(λu2(β)+(1λ)u2+(β))(λv2(β)+(1λ)v2+(β))},supβrmaxλ[0,1]{(λu2(β)+(1λ)u2+(β))(λv2(β)+(1λ)v2+(β))},
    for any r[0,1].

  3. Multiplication of the fuzzy regular prismoid numbers satisfies

    [u~υ~]r=infβrminλ[0,1]{(λu1(β)+(1λ)u1+(β))(λv1(β)+(1λ)v1+(β))},supβrmaxλ[0,1]{(λu1(β)+(1λ)u1+(β))(λv1(β)+(1λ)v1+(β))}×infβrminλ[0,1]{(λu2(β)+(1λ)u2+(β))(λv2(β)+(1λ)v2+(β))},supβrmaxλ[0,1]{(λu2(β)+(1λ)u2+(β))(λv2(β)+(1λ)v2+(β))},
    for any r[0,1].

  4. Division of the fuzzy regular prismoid numbers satisfies

    [u~υ~]r=infβrminλ[0,1]{(λu1(β)+(1λ)u1+(β))÷(λv1(β)+(1λ)v1+(β))},supβrmaxλ[0,1]{(λu1(β)+(1λ)u1+(β))÷(λv1(β)+(1λ)v1+(β))}×infβrminλ[0,1]{(λu2(β)+(1λ)u2+(β))÷(λv2(β)+(1λ)v2+(β))},supβrmaxλ[0,1]{(λu2(β)+(1λ)u2+(β))÷(λv2(β)+(1λ)v2+(β))},
    for any r[0,1].

Proposition 1.

Let ũ=(a1,a2,a3,a4;b1,b2,b3,b4),ṽ=(c1,c2,c3,c4;d1,d2,d3,d4) and w̃=(e1,e2,e3,e4;f1,f2,f3,f4) be all fuzzy regular prismoid numbers. Then

  1. For any r[0,1],

    [u~v~]r=[(a2a1+c2c1)r+a1+c1,a4+c4(a4a3+c4c3)r]×[(b2b1+d2d1)r+b1+d1,b4+d4(b4b3+d4d3)r];

  2. For any r[0,1],

    [u~υ~]r=infβrmin(a2a1c2+c1)β+a1c1,a4c4(a4a3c4+c3)β,supβrmax(a2a1c2+c1)β+a1c1,a4c4(a4a3c4+c3)β×infβrmin(b2b1d2+d1)β+b1d1,b4d4(b4b3d4+d3)β,supβrmax(b2b1d2+d1)β+b1d1,b4d4(b4b3d4+d3)β;

  3. ũũ={(0,0)};

  4. (ũṽ)ṽ=ũ;

  5. ũ(ṽw̃)=(ũṽ)(ũw̃);

  6. ũ(ṽw̃)=(ũṽ)(ũw̃);

  7. ũũ={(1,1)};

  8. (ũṽ)ṽ=ũ;

  9. ũ(ṽw̃)=(ũṽ)(ũw̃);

  10. ũ(ṽw̃)=(ũṽ)(ũw̃).

Proof.

According to Definition 2, the proof of (1), (2), (3) and (7) are immediate.

(4) We have from (1) and (2) that

[(u~υ~)υ~]r=infβrmin(a2a1)β+a1,a4(a4a3)β,supβrmax(a2a1)β+a1,a4(a4a3)β×infβrmin(b2b1)β+b1,b4(b4b3)β,supβrmax(b2b1)β+b1,b4(b4b3)β=(a2a1)r+a1,a4(a4a3)r×(b2b1)r+b1,b4(b4b3)r=[u~]r,
for any r[0,1], which implies that (ũṽ)ṽ=ũ.

(5) It follows from Definition 2 and (1) that

u~(v~w~)r=infβrminλ[0,1]λ(a2a1)β+a1+(1λ)a4(a4a3)βλ(c2c1+e2e1)β+c1+e1+(1λ)c4+e4(c4c3+e4e3)β,supβrmaxλ[0,1]λ(a2a1)β+a1+(1λ)a4(a4a3)βλ(c2c1+e2e1)β+c1+e1+(1λ)c4+e4(c4c3+e4e3)β]×infβrminλ[0,1]λ(b2b1)β+b1+(1λ)b4(b4b3)βλ(d2d1+f2f1)β+d1+f1+(1λ)d4+f4(d4d3+f4f3)β,supβrmaxλ[0,1]λ(b2b1)β+b1+(1λ)b4(b4b3)βλ(d2d1+f2f1)β+d1+f1+(1λ)d4+f4(d4d3+f4f3)β,
for any r[0,1]. According to the law of the multiplication allocating rule, we obtain
[u~(υ~w~)]r=[infβrminλ[0,1]λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β]λ[(c2c1)β+c1]+(1λ)[c4(c4c3)β]+λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β]λ[(e2e1)β+e1]+(1λ)[e4(e4e3)β],supβrmaxλ[0,1]λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β]λ[(c2c1)β+c1]+(1λ)[c4(c4c3)β]+λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β]λ[(e2e1)β+e1]+(1λ)[e4(e4e3)β]×[infβrminλ[0,1]λ[(b2b1)β+b1]+(1λ)[b4(b4b3)β]λ[(d2d1)β+d1]+(1λ)[d4(d4d3)β]+λ[(b2b1)β+b1]+(1λ)[b4(b4b3)β]λ[(f2f1)β+f1]+(1λ)[f4(f4f3)β],supβrmaxλ[0,1]λ[(b2b1)β+b1]+(1λ)[b4(b4b3)β]λ[(d2d1)β+d1]+(1λ)[d4(d4d3)β]+λ[(b2b1)β+b1]+(1λ)b4(b4b3)βλ[(f2f1)β+f1]+(1λ)[f4(f4f3)β],
for any r[0,1]. Based on the monotonicity of f1(λ)=λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β],f2(λ)=λ[(b2b1)β+b1]+(1λ)[b4(b4b3)β],f3(λ)=λ[(c2c1)β+c1]+(1λ)[c4(c4c3)β],f4(λ)=λ[(d2d1)β+d1]+(1λ)[d4(d4d3)β],f5(λ)=λ[(e2e1)β+e1]+(1λ)[e4(e4e3)β] and f6(λ)=λ[(f2f1)β+f1]+(1λ)[f4(f4f3)β], we have
[u~(υ~w~)]r=infβrminλ[0,1](λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β])(λ[(c2c1)β+c1]+(1λ)[c4(c4c3)β])+infβrminλ[0,1]λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β]λ[(e2e1)β+e1]+(1λ)[+e4(e4e3)β],supβrmaxλ[0,1]λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β]λ[(c2c1)β+c1]+(1λ)[c4(c4c3)β]+supβrmaxλ[0,1]λ[(a2a1)β+a1]+(1λ)[a4(a4a3)β]λ[(e2e1)β+e1]+(1λ)[e4(e4e3)β]×infβrminλ[0,1]λ[(b2b1)β+b1]+(1λ)[b4(b4b3)β]λ[(d2d1)β+d1]+(1λ)[d4(d4d3)β]+infβrminλ[0,1]λ[(b2b1)β+b1]+(1λ)[b4(b4b3)β]λ[(f2f1)β+f1]+(1λ)[f4(f4f3)β],supβrmaxλ[0,1]λ[(b2b1)β+b1]+(1λ)[b4(b4b3)β]λ[(d2d1)β+d1]+(1λ)[d4(d4d3)β]+supβrmaxλ[0,1]λ[(b2b1)β+b1]+(1λ)[b4(b4b3)β]λ[(f2f1)β+f1]+(1λ)[f4(f4f3)β],
for any r[0,1]. Then,
[ũ(ṽw̃)]r=[(ũṽ)(ũw̃)]r,
for any r[0,1], which implies that ũ(ṽw̃)=(ũṽ)(ũw̃).

The proof of (8) similar to (4) as well as the proof of (6), (9) and (10) are similar to (5).

2.2. Fuzzy Regular Prismoid Numbers Approximation of 2-Dimensional Fuzzy Numbers

The fuzzy regular prismoid numbers approximation of 2-dimensional fuzzy numbers will be discussed, which is an indispensable part to study the fuzzy risk analysis based on fuzzy regular prismoid numbers.

Lemma 1.

[46] Let f:RnR be a strictly convex and differentiable function, g1,g2,,gm:RnR be convex and differentiable functions. Then x¯ solves the convex programing problem

minf(x),

s.t.gi(x)bi(i=1,2,,m),

if and only if there exist μi(i=1,2,,m), such that

  1. f(x¯)+Σi=1mμigi(x¯)=0,

  2. gi(x¯)bi0,

  3. μi0,

  4. μi(bigi(x¯))=0.

Let ũE2. We will try to find a fuzzy regular prismoid number ṽP(E2), which is the nearest fuzzy regular prismoid number of ũ and preserves the core of ũ with respect to the weighted pseudometric. Thus we have to find such real numbers a1,a2,a3,a4,b1,b2,b3 and b4 that minimize

D(u~,υ~)=12{01f(r)(Rp̃u(x)1(r)a4+(a4a3)r)2dr+01f(r)(Rp̃u(y)1(r)b4+(b4b3)r)2dr12+01f(r)(Lp̃u(x)1(r)a1(a2a1)r)2dr+01f(r)(Lp̃u(y)1(r)b1(b2b1)r)2dr12,
with respect to condition coreũ=coreṽ, i.e.,
a2=Lp̃u(x)1(1),a3=Rp̃u(x)1(1),b2=Lp̃u(y)1(1),b3=Rp̃u(y)1(1).

Therefore, we only need minimize the function

F(a1,a4,b1,b4)=01f(r)Lp~u(x)1(r)a1(Lp~u(x)1(1)a1)r2dr+01f(r)Rp~u(x)1(r)a4+(a4Rp~u(x)1(1))r)2dr+01f(r)Lp~u(y)1(r)b1(Lp~u(y)1(1)b1)r)2dr+01f(r)Rp~u(y)1(r)b4+(b4Rp~u(y)1(1))r)2dr=(a1)201f(r)(1r)2dr+(a4)201f(r)(1r)2dr+(b1)201f(r)(1r)2dr+(b4)201f(r)(1r)2dr+2a101f(r)(1r)rLp~u(x)1(1)Lp~u(x)1(r)dr+2a401f(r)(1r)rRp~u(x)1(1)Rp~u(x)1(r)dr+2b101f(r)(1r)rLp~u(x)1(1)Lp~u(x)1(r)dr+2b401f(r)(1r)rRp~u(x)1(1)Rp~u(x)1(r)dr+01f(r)(Lp~u(x)1(r)rLp~u(x)1(1))2dr+01f(r)(Rp~u(x)1(r)rRp~u(x)1(1))2dr+01f(r)(Lp~u(y)1(r)rLp~u(y)1(1))2dr+01f(r)(Rp~u(y)1(r)rRp~u(y)1(1))2dr,(1)
subject to
a1a40,b1b40.(2)

Theorem 1.

Suppose that ũE2 and the membership function for the projective of ũ on the xoz and yoz plane satisfy

p̃u(x)={Lx(x),axx<bx,1,bxxcx,Rx(x),cx<xdx,0,otherwise,
p̃u(y)={Ly(y),ayy<by,1,byycy,Ry(y),cy<ydy,0,otherwise,
respectively. Furthermore, ṽP(E2) and
υ̃(x,y)={xa1a2a1,a1xa2,(b2b1)x+a2b1a1b2a2a1y(b3b4)x+a2b4a1b3a2a1,yb1b2b1,b1yb2,(a2a1)y+a1b2a2b1b2b1x(a3a4)y+a4b2a3b1b2b1,1,a2xa3,b2yb3,a4xa4a3,a3xa4,(b2b1)x+a3b1a4b2a3a4y(b3b4)x+a3b4a4b3a3a4,b4yb4b3,b3yb4,(a2a1)y+a1b3a2b4b3b4x(a3a4)y+a4b3a3b4b3b4,0,otherwise.

ṽ is the nearest fuzzy regular prismoid number to ũ preserves the core of ũ with respect to the weighted pseudometric.

  1. If

    01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr>01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,
    01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr>01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr,
    we have
    a1=a4=01f(r)(1r)[(Lp̃u(x)1(r)rLp̃u(x)1(1))+(Rp̃u(x)1(r)rRp̃u(x)1(1))]dr201f(r)(1r)2dr,
    b1=b4=01f(r)(1r)[(Lp̃u(y)1(r)rLp̃u(y)1(1))+(Rp̃u(y)1(r)rRp̃u(y)1(1))]dr201f(r)(1r)2dr.

  2. If

    01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr>01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,
    01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr,
    we have
    a1=a4=01f(r)(1r)[(Lp̃u(x)1(r)rLp̃u(x)1(1))+(Rp̃u(x)1(r)rRp̃u(x)1(1))]dr201f(r)(1r)2dr,
    b1=01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)2dr,
    b4=01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr01f(r)(1r)2dr.

  3. If

    01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,
    01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr>01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr,
    we have
    a1=01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)2dr,
    a4=01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr01f(r)(1r)2dr,
    b1=b4=01f(r)(1r)[(Lp̃u(y)1(r)rLp̃u(y)1(1))+(Rp̃u(y)1(r)rRp̃u(y)1(1))]dr201f(r)(1r)2dr.

  4. If

    01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,
    01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr,
    we have
    a1=01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)2dr,
    a4=01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr01f(r)(1r)2dr,
    b1=01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)2dr,
    b4=01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr01f(r)(1r)2dr.

    Where the function f(r) is nonnegative and increasing on [0,1] with f(0)=0, 01f(r)dr=12.

Proof.

Because the function F in (1) and the conditions (2) satisfy the hypothesis of convexity and differentiability in Lemma 1, the conditions (i)–(iv) in Lemma 1 with respect to the minimization problem (1) in the conditions (2) can be shown as following:

   2a101f(r)(1r)2dr+201f(r)(1r)rLp̃u(x)1(1)Lp̃u(x)1(r)dr+μ1=0,(3)
   2a401f(r)(1r)2dr+201f(r)(1r)rRp̃u(x)1(1)Rp̃u(x)1(r)drμ1=0,(4)
   2b101f(r)(1r)2dr+201f(r)(1r)rLp̃u(y)1(1)Lp̃u(y)1(r)dr+μ2=0,(5)
   2b401f(r)(1r)2dr+201f(r)(1r)rRp~u(y)1(1)Rp~u(y)1(r)drμ2=0,(6)
μ1(a1a4)=0,(7)
μ2(b1b4)=0,(8)
μ10,(9)
μ20.(10)
  1. In the case μ1>0 and μ2>0, the solution of the system (3)(10) is

    a1=a4=01f(r)(1r)[(Lp̃u(x)1(r)rLp̃u(x)1(1))+(Rp̃u(x)1(r)rRp̃u(x)1(1))]dr201f(r)(1r)2dr,
    b1=b4=01f(r)(1r)[(Lp̃u(y)1(r)rLp̃u(y)1(1))+(Rp̃u(y)1(r)rRp̃u(y)1(1))]dr201f(r)(1r)2dr,
    μ1=01f(r)(1r)[(Lp̃u(x)1(r)rLp̃u(x)1(1))(Rp̃u(x)1(r)rRp̃u(x)1(1))]dr,
    μ2=01f(r)(1r)[(Lp̃u(y)1(r)rLp̃u(y)1(1))(Rp̃u(y)1(r)rRp̃u(y)1(1))]dr.

    It is obvious that μ1>0 and μ2>0. Then the conditions (3)(10) are verified. Furthermore, we can prove that

    a4a3=01f(r)(1r)[(Lp̃u(x)1(r)rLp̃u(x)1(1))+(Rp̃u(x)1(r)rRp̃u(x)1(1))]dr201f(r)(1r)2drRp̃u(x)1(1)01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr01f(r)(1r)2drRp̃u(x)1(1)=01f(r)(1r)(Rp̃u(x)1(r)Rp̃u(x)1(1))dr01f(r)(1r)2dr0,
    a2a1=Lp̃u(x)1(1)01f(r)(1r)[(Lp̃u(x)1(r)rLp̃u(x)1(1))+(Rp̃u(x)1(r)rRp̃u(x)1(1))]dr201f(r)(1r)2drLp̃u(x)1(1)01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)2dr=01f(r)(1r)(Lp̃u(x)1(1)Lp̃u(x)1(r))dr01f(r)(1r)2dr0.

    Which implies that a4a3a2a1. Similarly, we have b4b3b2b1. Then ṽ=(a1,a2,a3,a4;b1,b2,b3,b4) is the nearest fuzzy regular prismoid number to ũ in this case.

  2. In this case μ1>0 and μ2=0, the solution of the system (3)(10) is

    a1=a4=01f(r)(1r)[(Lp̃u(x)1(r)rLp̃u(x)1(1))+(Rp̃u(x)1(r)rRp̃u(x)1(1))]dr201f(r)(1r)2dr,
    b1=01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)2dr,
    b4=01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr01f(r)(1r)2dr,
    μ1=01f(r)(1r)[(Lp̃u(x)1(r)rLp̃u(x)1(1))(Rp̃u(x)1(r)rRp̃u(x)1(1))]dr,
    μ2=0.

    It is obvious that μ1>0 and μ2=0. Then the conditions (3)(10) are verified. Similar to (1) we have a4a3a2a1. Moreover, we can prove that

    b4b3=01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr01f(r)(1r)2drRp̃u(y)1(1)=01f(r)(1r)(Rp̃u(y)1(r)Rp̃u(y)1(1))dr01f(r)(1r)2dr0,
    b2b1=Lp̃u(y)1(1)01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)2dr=01f(r)(1r)(Lp̃u(y)1(1)Lp̃u(y)1(r))dr01f(r)(1r)2dr0.

    Which implies that b4b3b2b1. Then ṽ=(a1,a2,a3,a4;b1,b2,b3,b4) is the nearest fuzzy regular prismoid number to ũ in this case.

  3. In the case μ1=0 and μ2>0, the solution of the system (3)(10) is

    a1=01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)2dr,
    a4=01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr01f(r)(1r)2dr,
    b1=b4=01f(r)(1r)[(Lp̃u(y)1(r)rLp̃u(y)1(1))+(Rp̃u(y)1(r)rRp̃u(y)1(1))]dr201f(r)(1r)2dr,
    μ1=0,
    μ2=01f(r)(1r)[(Lp̃u(y)1(r)rLp̃u(y)1(1))(Rp̃u(y)1(r)rRp̃u(y)1(1))]dr.

    It is obvious that μ1=0 and μ2>0. Then the conditions (3)(10) are verified. Similar to (1) and (2) we have a4a3a2a1 and b4b3b2b1. Then ṽ=(a1,a2,a3,a4;b1,b2,b3,b4) is the nearest fuzzy regular prismoid number to ũ in this case.

  4. In the case μ1=0 and μ2=0, the solution of the system (3)(10) is

    a1=01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)2dr,
    a4=01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr01f(r)(1r)2dr,
    b1=01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)2dr,
    b4=01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr01f(r)(1r)2dr,
    μ1=0,
    μ2=0.

Then the conditions (3)(10) are verified. Similar to (1) and (2) we have a4a3a2a1 and b4b3b2b1. Then ṽ=(a1,a2,a3,a4;b1,b2,b3,b4) is the nearest fuzzy regular prismoid number to ũ in this case.

For any 2-dimensional fuzzy number, we can apply one and only one of the above situations (1)(4) to calculate the fuzzy regular prismoid approximation of it. We denote

Ω1={ũE2:01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr>01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr>01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr},
Ω2={ũE2:01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr>01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr},
Ω3={ũE2:01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr>01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr},
Ω4={ũE2:01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr}.

It is obvious that the cases (1)(4) cover the set of all 2-dimensional fuzzy numbers and Ω1,Ω2,Ω3,Ω4 are disjoint sets. So the approximation operator always gives a fuzzy regular prismoid number.

Example 1.

Let ũE2. If the r-level set of ũ satisfies

[u~]r=0.0115r2+0.3479r+2.51220.26r+3.04,0.009r20.4136r+4.33264.410.29r×0.0141r2+0.4047r+2.80620.27r+3.53,0.0025r20.327r+4.49094.660.22r,
for any r[0,1], and [u]1[0.8702,0.9534]×[0.8487,0.9384]. It is obvious that the projective of ũ on the xoz and yoz plane satisfy
[p~u(x)]r=0.0115r2+0.3479r+2.51220.26r+3.04,0.009r20.4136r+4.33264.410.29r,
[p~u(y)]r=0.0141r2+0.4047r+2.80620.27r+3.53,0.0025r20.327r+4.49094.660.22r,
for any r[0,1] respectively. Taking f(r)=r, we have
01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr0.0689,
01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr0.0819,
01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr0.0662,
01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr0.0803.

Then,

01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr<01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr,
01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr<01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr.

It follows from Theorem 1 that

a1=01f(r)(1r)(Lp̃u(x)1(r)rLp̃u(x)1(1))dr01f(r)(1r)2dr0.8263,
a4=01f(r)(1r)(Rp̃u(x)1(r)rRp̃u(x)1(1))dr01f(r)(1r)2dr0.9824,
b1=01f(r)(1r)(Lp̃u(y)1(r)rLp̃u(y)1(1))dr01f(r)(1r)2dr0.7950,
b4=01f(r)(1r)(Rp̃u(y)1(r)rRp̃u(y)1(1))dr01f(r)(1r)2dr0.9640.

Therefore, the fuzzy regular prismoid number ṽ=(0.8263,0.8702,0.9534,0.9824;0.7950,0.8487,0.9384,0.9640) is the nearest fuzzy regular prismoid number to ũ preserves the core of ũ with respect to the weighted distance.

3. THE SIMILARITY MEASURE BETWEEN FUZZY REGULAR PRISMOID NUMBERS

3.1. The Radius of Gyration for Fuzzy Regular Prismoid Numbers

We will mention the moment of inertia and the radius of gyration about a region of space Ω. Consider a space Ω located in rectangular coordinate system Oxyz. The moment of inertia for the space Ω with respect to the x-axis, the moment of inertia for the space Ω with respect to the y-axis and the moment of inertia for the space Ω with respect to the z-axis are defined, respectively, as

Ix=Ω(y2+z2)dV,(11)
Iy=Ω(z2+x2)dV,(12)
Iz=Ω(x2+y2)dV.(13)

The radius of gyration for a space Ω with respect to x-axis, y-axis and z-axis are defined, respectively, as

rx=IxV,(14)
ry=IyV,(15)
rz=IzV,(16)
where V is the volume of the space Ω.

Example 2.

For fuzzy regular prismoid number ũ=(0.72,0.78,0.92,0.97;0.72,0.78,0.92,0.97) with membership function as given in Definition 1.

Let Ω1={(x,y,z):0.72x0.78,xy0.83x+1.57,0z16.67x12}.

Ω1(y2+z2)dV=0.720.78dxx0.83x+1.57dy016.67x12(y2+z2)dz0.0046,
Ω1(z2+x2)dV=0.720.78dxx0.83x+1.57dy016.67x12(z2+x2)dz0.0039,
Ω1(x2+y2)dV=0.720.78dxx0.83x+1.57dy016.67x12(x2+y2)dz0.0069,
Ω11dV=0.720.78dxx0.83x+1.57dy016.67x121dz0.0053.

Let Ω2={(x,y,z):0.72y0.78,yx0.83y+1.57,0z16.67y12}.

Ω2(y2+z2)dV=0.720.78dyy0.83y+1.57dx016.67y12(y2+z2)dz0.0039,
Ω2(z2+x2)dV=0.720.78dyy0.83y+1.57dx016.67y12(z2+x2)dz0.0046,
Ω2(x2+y2)dV=0.720.78dyy0.83y+1.57dx016.67y12(x2+y2)dz0.0069,
Ω21dV=0.720.78dyy0.83y+1.57dx016.67y121dz0.0053.

Let Ω3={(x,y,z):0.92x0.97,1.2x+1.884yx,0z20x+19.4}.

Ω3(y2+z2)dV=0.920.97dx1.2x+1.884xdy020x+19.4(y2+z2)dz0.0039,
Ω3(z2+x2)=0.920.97dx1.2x+1.884xdy020x+19.4(z2+x2)dz0.0046,
Ω3(x2+y2)=0.920.97dx1.2x+1.884xdy020x+19.4(x2+y2)dz0.0071,
Ω31dV=0.920.97dx1.2x+1.884xdy020x+19.41dz0.0044.

Let Ω4={(x,y,z):0.92y0.97,1.2y+1.884xy,0z20y+19.4}.

Ω4(y2+z2)dV=0.720.78dyy0.83y+1.57dx016.67y12(y2+z2)dz0.0046,
Ω4(z2+x2)dV=0.720.78dyy0.83y+1.57dx016.67y12(z2+x2)dz0.0039,
Ω4(x2+y2)dV=0.720.78dyy0.83y+1.57dx016.67y12(x2+y2)dz0.0071,
Ω41dV=0.720.78dyy0.83y+1.57dx016.67y121dz0.0044.

Let Ω5={(x,y,z):0.78x0.92,0.78y0.92,0z1}.

Ω5(y2+z2)dV=0.780.92dx0.780.92dy01(y2+z2)dz0.0207,
Ω5(z2+x2)dV=0.780.92dx0.780.92dy01(z2+x2)dz0.0207,
Ω5(x2+y2)dV=0.780.92dx0.780.92dy01(x2+y2)dz0.0284,
Ω51dV=0.780.92dx0.780.92dy011dz0.0196.

Let Ω=Ω1+Ω2+Ω3+Ω4+Ω5. We have from (11), (12) and (13) that

Ix=Ω(y2+z2)dV=Ω1(y2+z2)dV+Ω2(y2+z2)dV+Ω3(y2+z2)dV+Ω4(y2+z2)dV+Ω5(y2+z2)dV0.0376,
Iy=Ω(z2+x2)dV=Ω1(z2+x2)dV+Ω2(z2+x2)dV+Ω3(z2+x2)dV+Ω4(z2+x2)dV+Ω5(z2+x2)dV0.0376,
Iz=Ω(x2+y2)dV=Ω1(x2+y2)dV+Ω2(x2+y2)dV+Ω3(x2+y2)dV+Ω4(x2+y2)dV+Ω5(x2+y2)dV0.0563.

Furthermore, we have

V=Ω1dV=Ω11dV+Ω21dV+Ω31dV+Ω41dV+Ω51dV0.0390.

According to (14), (15) and (16), we obtain the radius of gyration for fuzzy regular prismoid number ũ with respect to x-axis, y-axis and z-axis are rx=IxV0.9821,ry=IyV0.9821,rz=IzV1.2009, respectively.

3.2. The radius of gyration similarity measure between fuzzy regular prismoid numbers

A similarity measure for fuzzy regular prismoid numbers is being proposed, which is based on distance and the radius of gyration for fuzzy regular prismoid numbers.

Definition 3.

Let ũ and ṽ be both fuzzy regular prismoid numbers. The degree of similarity between the fuzzy regular prismoid numbers ũ and ṽ, denoted as S(ũ,ṽ), is defined as

S(ũ,ṽ)=11+D(ũ,ṽ)×min{rx(ũ),rx(ṽ)}+min{ry(ũ),ry(ṽ)}+min{rz(ũ),rz(ṽ)}max{rx(ũ),rx(ṽ)}+max{ry(ũ),ry(ṽ)}+max{rz(ũ),rz(ṽ)},
where, rx(ũ),ry(ũ) and rz(ũ) are the radius of gyration for ũ with respect to x-axis, y-axis and z-axis respectively, as well as rx(ṽ),ry(ṽ) and rz(ṽ) are the radius of gyration for ṽ with respect to x-axis, y-axis and z-axis respectively.

The larger value of S(ũ,ṽ) gives the more similarity between the fuzzy regular prismoid numbers ũ and ṽ.

Some of the properties for the similarity measure of fuzzy regular prismoid numbers are researched as follows.

Property 2.

Let ũ=(a1,a2,a3,a4;b1,b2,b3,b4) and ṽ=(c1,c2,c3,c4;d1,d2,d3,d4) be both fuzzy regular prismoid numbers. The degree of similarity S(ũ,ṽ)=1 if and only if fuzzy regular prismoid numbers ũ and ṽ are identical.

Proof.

If fuzzy regular prismoid numbers ũ and ṽ are identical, i.e. a1=c1,a2=c2,a3=c3,a4=c4,b1=d1,b2=d2,b3=d3,b4=d4. It is obvious that D(ũ,ṽ)=0,rx(ũ)=rx(ṽ),ry(ũ)=ry(ṽ) and rz(ũ)=rz(ṽ), thus

S(ũ,ṽ)=11+D(ũ,ṽ)×min{rx(ũ),rx(ṽ)}+min{ry(ũ),ry(ṽ)}+min{rz(ũ),rz(ṽ)}max{rx(ũ),rx(ṽ)}+max{ry(ũ),ry(ṽ)}+max{rz(ũ),rz(ṽ)}=1.

Only if: Assume that S(ũ,ṽ)=1. According to

11+D(ũ,ṽ)1,
min{rx(ũ),rx(ṽ)}+min{ry(ũ),ry(ṽ)}+min{rz(ũ),rz(ṽ)}max{rx(ũ),rx(ṽ)}+max{ry(ũ),ry(ṽ)}+max{rz(ũ),rz(ṽ)}1,
we have D(ũ,ṽ)=0 and
min{rx(ũ),rx(ṽ)}+min{ry(ũ),ry(ṽ)}+min{rz(ũ),rz(ṽ)}max{rx(ũ),rx(ṽ)}+max{ry(ũ),ry(ṽ)}+max{rz(ũ),rz(ṽ)}=1,
which implies that ũ=ṽ.

Property 3.

Let ũ,ṽP(E2). Then S(ũ,ṽ)=S(ṽ,ũ).

Proof.

We have from D(ũ,ṽ)=D(ṽ,ũ),min{rx(ũ),rx(ṽ)}=min{rx(ṽ),rx(ũ)},min{ry(ũ),ry(ṽ)}=min{ry(ṽ),ry(ũ)},min{rz(ũ),rz(ṽ)}=min{rz(ṽ),rz(ũ)},max{rx(ũ),rx(ṽ)}=max{rx(ṽ),rx(ũ)},max{ry(ũ),ry(ṽ)}=max{ry(ṽ),ry(ũ)},max{rz(ũ),rz(ṽ)}=max{rz(ṽ),rz(ũ)} that

S(ũ,ṽ)=11+D(ũ,ṽ)×min{rx(ũ),rx(ṽ)}+min{ry(ũ),ry(ṽ)}+min{rz(ũ),rz(ṽ)}max{rx(ũ),rx(ṽ)}+max{ry(ũ),ry(ṽ)}+max{rz(ũ),rz(ṽ)}=11+D(ṽ,ũ)×min{rx(ṽ),rx(ũ)}+min{ry(ṽ),ry(ũ)}+min{rz(ṽ),rz(ũ)}max{rx(ṽ),rx(ũ)}+max{ry(ṽ),ry(ũ)}+max{rz(ṽ),rz(ũ)}=S(ṽ,ũ).

Property 4.

Let ũ,ṽP(E2). Then 0S(ũ,ṽ)1.

Proof.

From 011+D(ũ,ṽ)1 and min{rx(ũ),rx(ṽ)}+min{ry(ũ),ry(ṽ)}+min{rz(ũ),rz(ṽ)}max{rx(ũ),rx(ṽ)}+max{ry(ũ),ry(ṽ)}+max{rz(ũ),rz(ṽ)}, we have 0S(ũ,ṽ)1.

Example 3.

For fuzzy regular prismoid numbers ũ=(0.72,0.78,0.92,0.97;0.72,0.78,0.92,0.97) and ṽ=(0.8263,0.8702,0.9534,0.9824;0.7950,0.8487,0.9384,0.9640) with membership function as given in Definition 1.

According to Example 2, the radius of gyration for fuzzy regular prismoid number ũ with respect to x-axis, y-axis and z-axis are rx(ũ)=IxV0.9821,ry(ũ)=IyV0.9821,rz(ũ)=IzV1.2009.

For ṽ=(0.8341,0.8701,0.9535,0.9895;0.8086,0.8488,0.9382,0.9784), we have Ix=Ω(y2+z2)dV0.0165,Iy=Ω(z2+x2)dV0.0171,Iz=Ω(x2+y2)dV0.0261 and V=Ω1dV0.0160. According to (14), (15) and (16), we obtain the radius of gyration for fuzzy regular prismoid number ṽ with respect to x-axis, y-axis and z-axis are rx(ṽ)=IxV1.0108,ry(ṽ)=IyV1.0301,rz(ṽ)=IzV1.2688, respectively. From

D(u~,υ~)=1201f(r)(Rp̃u(x)1(r)Rp̃v(x)1(r))2dr+01f(r)(Rp̃u(y)1(r)Rp̃v(y)1(r))2dr12+01f(r)(Lp̃u(x)1(r)Lp̃v(x)1(r))2dr+01f(r)(Lp̃u(y)1(r)Lp̃v(y)1(r))2dr12=1201f(r)(a4c4(a4c4a3+c3)r)2dr+01f(r)(b4d4(b4d4b3+d3)r)2dr12+01f(r)((a2a1c2+c1)r+a1c1)2dr+01f(r)((b2b1d2+d1)r+b1d1)2dr120.0524,
we have
S(ũ,ṽ)=11+D(ũ,ṽ)×min{rx(ũ),rx(ṽ)}+min{ry(ũ),ry(ṽ)}+min{rz(ũ),rz(ṽ)}max{rx(ũ),rx(ṽ)}+max{ry(ũ),ry(ṽ)}+max{rz(ũ),rz(ṽ)}11.0524×min{0.9821,1.0108}+min{0.9821,1.0301}+min{1.2009,1.2688}max{0.9821,1.0108}+max{0.9821,1.0301}+max{1.2009,1.2688}=11.0524×0.9821+0.9821+1.20091.0108+1.0301+1.26880.9087.

4. Fuzzy risk analysis based on fuzzy regular prismoid numbers

Assume that there is a component A consisting of n sub-components A1,A2,,An and each sub-component is evaluated by two evaluating items probability of failure and severity of loss, as well as R̃i denotes the probability of failure and ω̃i denotes the severity of loss of the sub-component Ai, respectively, where R̃i and ω̃i are regular prismoid numbers (1in). The algorithm for dealing with fuzzy risk analysis is now presented as follows:

Step 1: Use the regular prismoid number arithmetic operations to calculate the probability of failure R̃ of component A,

   R~=i=1nR~iω~ii=1nω~i,
where R̃E2.

Step 2: Use Theorem 1 to calculate the nearest fuzzy regular prismoid number R̃ to R̃ preserves the centroid of [R̃]1 with respect to the weighted pseudometric.

Step 3: Calculate the radius of gyration for each linguistic term shown in Table 1 with respect to x-axis, y-axis and z-axis.

Linguistic Terms Linguistic Valued Regular Prismoid Number
Absolutely low (0, 0, 0, 0; 0, 0, 0, 0)
Very low (0, 0, 0.02, 0.07; 0, 0, 0.02, 0.07)
Low (0.04, 0.1, 0.18, 0.23; 0.04, 0.1, 0.18, 0.23)
Fairly low (0.17, 0.22, 0.36, 0.42; 0.17, 0.22, 0.36, 0.42)
Medium (0.32, 0.41, 0.58, 0.65; 0.32, 0.41, 0.58, 0.65)
Fairly high (0.58, 0.63, 0.80, 0.86; 0.58, 0.63, 0.80, 0.86)
High (0.72, 0.78, 0.92, 0.97; 0.72, 0.78, 0.92, 0.97)
Very high (0.93, 0.98, 1.0, 1.0; 0.93, 0.98, 1.0, 1.0)
Absolutely high (1.0, 1.0, 1.0, 1.0; 1.0, 1.0, 1.0, 1.0)
Table 1

A 9-member linguistic term set.

Step 4: According to the distance D of the fuzzy regular prismoid numbers, we calculate the distance of R̃ with each linguistic term.

Step 5: Use the similarity measure of regular prismoid numbers to calculate the similarity measure of R̃ with each linguistic term.

Step 6: Find the largest similarity of R̃ and it is considered as a risk value of the system in linguistic term. The largest the similarity measure, the highest the probability of failure of component A.

Example 4.

Every company plays a major role in creating employment opportunities and improving the standard of living of people. Therefore, it is necessary to study the fuzzy risk analysis on every company. The key to success of a company is to depend on the following sub-components. A1: Business management, which based on the personal ability and management inexperience. A2: Working efficiency of employee, which based on their production speed and the quality of their products. A3: Technology, which based on the hardware investments and the technological level of employees. A4: Product, which based on products popularity and products market. A5: Geographical location, which based on transportation and peripheral economic. Obviously, these linguistic terms are expressed by fuzzy regular prismoid numbers more precise and more actual. Assume that the fuzzy regular prismoid numbers of the probability of failure and severity of loss of the sub-components are shown in Table 2.

Ai Probability of Failure R̃i Severity of Loss ω̃i
A1 (0.93,0.98,1.0,1.0;0.93,0.98,1.0,1.0) (0.58,0.63,0.80,0.86;0.58,0.63,0.80,0.86)
A2 (0.72,0.78,0.92,0.97;0.93,0.98,1.0,1.0) (0.58,0.63,0.80,0.86;0.58,0.63,0.80,0.86)
A3 (1.0,1.0,1.0,1.0;0.72,0.78,0.92,0.97) (0.72,0.78,0.92,0.97;0.72,0.78,0.92,0.97)
A4 (0.72,0.78,0.92,0.97;0.58,0.63,0.80,0.86) (0.58,0.63,0.80,0.86;0.93,0.98,1.0,1.0)
A5 (0.72,0.78,0.92,0.97;0.93,0.98,1.0,1.0) (0.58,0.63,0.80,0.86;0.72,0.78,0.92,0.97)
Table 2

Fuzzy regular prismoid numbers of R~i and ω~i for Ssb-components A1; A2, …, A5.

Step 1: We have from Table 2 that [R̃1]r=[0.05r+0.93,1.0]×[0.05r+0.93,1.0], [ω̃1]r=[0.05r+0.58,0.860.06r]×[0.05r+0.58,0.860.06r], [R̃2]r=[0.06r+0.72,0.970.05r]×[0.05r+0.93,1.0], [ω̃2]r=[0.05r+0.58,0.860.06r]×[0.05r+0.58,0.860.06r], [R̃3]r=[1.0,1.0]×[0.06r+0.72,0.970.05r], [ω̃3]r=[0.06r+0.72,0.970.05r]×[0.06r+0.72,0.970.05r], [R̃4]r=[0.06r+0.72,0.970.05r]×[0.05r+0.58,0.860.06r], [ω̃4]r=[0.05r+0.58,0.860.06r]×[0.05r+0.93,1], [R̃5]r=[0.06r+0.72,0.970.05r]×[0.05r+0.93,1], [ω̃5]r=[0.05r+0.58,0.860.06r]×[0.06r+0.72,0.970.05r]. Then it follows from Definition 2 that

[R~1ω~1]r=infβrminλ[0,1]λ(0.05β+0.93)+(1λ)1.0λ(0.05β+0.58)+(1λ)(0.860.06β),supβrmaxλ[0,1]λ(0.05β+0.93)+(1λ)1.0λ(0.05β+0.58)+(1λ)(0.860.06β)×infβrminλ[0,1]λ(0.05β+0.93)+(1λ)1.0λ(0.05β+0.58)+(1λ)(0.860.06β),supβrmaxλ[0,1]λ(0.05β+0.93)+(1λ)1.0λ(0.05β+0.58)+(1λ)(0.860.06β)=infβr0.0025β2+0.0755β+0.5394,supβr{0.860.06β}×infβr{0.0025β2+0.0755β+0.5394},supβr{0.860.06β}=0.0025r2+0.0755r+0.5394,0.860.06r]×[0.0025r2+0.0755r+0.5394,0.860.06r,
for any r(0,1]. Similarly, we obtain
[R̃2ω̃2]r=[0.003r2+0.0708r+0.4176,0.003r20.1012r+0.8342]×[0.0025r2+0.0755r+0.5394,0.860.06r],[R̃3ω̃3]r=[0.003r2+0.0708r+0.4176,0.003r20.1012r+0.8342]×[0.0036r2+0.0864r+0.5184,0.0025r20.097r+0.9409],[R̃4ω̃4]r=[0.003r2+0.0708r+0.4176,0.003r20.1012r+0.8342]×[0.0025r2+0.0755r+0.5394,0.860.06r],[R̃5ω̃5]r=[0.003r2+0.0708r+0.4176,0.003r20.1012r+0.8342]×[0.003r2+0.0918r+0.6696,0.970.05r],
for any r(0,1]. Thus,
i=15Ri~ωi~r=[0.0115r2+0.3479r+2.5122,0.009r20.4136r+4.3326]×[0.0141r2+0.4047r+2.8062,0.0025r20.327r+4.4909]
for any r(0,1]. Furthermore, we have
i=15ω̃ir=[0.26r+3.04,4.410.29r]×[0.27r+3.53,4.660.22r],
for any r(0,1]. Then the probability of failure R̃ of component A is showing as follow.
[R̃]r=(i=1nR̃iω̃i)i=1nω̃ir=infβrminλ[0,1]λ(0.0115β2+0.3479β+2.5122)+(1λ)(0.009β20.4136β+4.3326)÷λ(0.26β+3.04)+(1λ)(4.410.29β),supβrmaxλ[0,1]λ(0.0115β2+0.3479β+2.5122)+(1λ)(0.009β20.4136β+4.3326)÷λ(0.26β+3.04)+(1λ)(4.410.29β)×infβrminλ[0,1]λ(0.0141β2+0.4047β+2.8062)+(1λ)(0.0025β20.327β+4.4909)÷{λ(0.27β+3.53)+(1λ)(4.660.22β)},supβrmaxλ[0,1]{λ(0.0141β2+0.4047β+2.8062)+(1λ)(0.0025β20.327β+4.4909)}÷λ(0.27β+3.53)+(1λ)(4.660.22β)=infβr0.0115β2+0.3479β+2.51220.26β+3.04,supβr0.009β20.4136β+4.33264.410.29β×infβr0.0141β2+0.4047β+2.80620.27β+3.53,supβr0.0025β20.327β+4.49094.660.22β=0.0115r2+0.3479r+2.51220.26r+3.04,0.009r20.4136r+4.33264.410.29r×0.0141r2+0.4047r+2.80620.27r+3.53,0.0025r20.327r+4.49094.660.22r,
for any r[0,1].

Step 2: According to Example 1 the nearest fuzzy regular prismoid number to R̃ preserves the centroid of [R̃]1 with respect to the weighted pseudometric is R̃=(0.8263,0.8702,0.9534,0.9824;0.7950,0.8487,0.9384,0.9640.;

Step 3: The radius of gyration for each linguistic term with respect to x-axis, y-axis and z-axis as shown in Table 3.

Linguistic Terms rx ry rz
Absolutelylow rx(AL)=0 ry(AL)=0 rz(AL)=0
Very-low rx(VL)0.4155 ry(VL)0.4155 rz(VL)0.0447
Low rx(L)0.4783 ry(L)0.4783 rz(AL)0.2033
Fairly-low rx(FL)0.5770 ry(FL)0.5770 rz(FL)0.4229
Medium rx(M)0.6911 ry(M)0.6911 rz(M)0.7003
Fairly high rx(FH)0.8802 ry(FH)0.8802 rz(FH)1.0198
High rx(H)0.9821 ry(H)0.9821 rz(H)1.2009
Very high rx(VH)1.0579 ry(VH)1.0579 rz(VH)1.3767
Absolutely high rx(AH)1.4142 ry(AH)1.4142 rz(AH)1.4142
Table 3

The radius of gyration for each linguistic term.

Step 4: The distance of R̃ with each linguistic term as shown in Table 4.

Linguistic Terms D
Absolutely low D(R̃,AL)0.8992
Very low D(R̃,VL)0.8809
Low D(R̃,L)0.7608
Fairly low D(R̃,FL)0.6075
Medium D(R̃,M)0.4077
Fairly high D(R̃,FH)0.1828
High D(R̃,H)0.0524
Very high D(R̃,VH)0.0835
Absolutely high D(R̃,AH)0.1019
Table 4

The distance of R~ with each linguistic term.

Step 5: Calculate the similarity measure of R̃ with each linguistic term, we have

S(R̃,Absolutelylow)=0
S(R̃,verylow)0.1407
S(R̃,low)0.1990
S(R̃,fairlylow)0.2964
S(R̃,medium)0.4470
S(R̃,fairlyhigh)0.7102
S(R̃,high)0.9087
S(R̃,veryhigh)0.8746
S(R̃,absolutelyhigh)0.7080

Step 6: It is obvious that 0.9087 is the largest value. Therefore, the the regular prismoid number R̃ is translated into the linguistic term high. That is, for this system the risk of failure to the company is high.

5. CONCLUSION

Taking into account that fuzzy regular prismoid number can provide more flexibility and tractability to represent the imprecise information, we will use the regular prismoid numbers to deal with fuzzy risk analysis problems. The arithmetic operator of fuzzy regular prismoid numbers and the degree of similarity between fuzzy regular prismoid numbers are introduced, which are the basic work of studying fuzzy risk analysis. Furthermore, we establish the fuzzy regular prismoid numbers approximation of 2-dimensional fuzzy numbers. Finally, the arithmetic operator and the degree of similarity for fuzzy regular prismoid numbers are applied in fuzzy risk analysis. In this sense the research is a first step toward solving fuzzy risk analysis. Our proposal for future work is to use fuzzy regular prismoid numbers to investigate fuzzy risk analysis and applications.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest

AUTHORS' CONTRIBUTIONS

From the conception of the idea, writing the manuscript, interpretation of the results to data preparation and analysis are all completed by the author independently.

Funding Statement

This work is supported by National Natural Science Foundation of China (11901265).

ACKNOWLEDGMENTS

The author thanks the editors, language editors and the anonymous reviewers for their suggestions, which improved this paper significantly.

REFERENCES

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14.D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, NY, USA, 1980. http://refhub.elsevier.com/S1568-4946(14)00547-X/sbref0015
20.M. Hukuhara, Integration des applications measurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj, Vol. 10, 1967, pp. 205-229. http://refhub.elsevier.com/S0165-0114(14)00155-9/bib68756Bs1
46.R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NY, USA, 1970. http://refhub.elsevier.com/S0165-0114(18)31080-7/bib5254526F636B6166656C6C617232303034s1
Journal
International Journal of Computational Intelligence Systems
Volume-Issue
14 - 1
Pages
1541 - 1563
Publication Date
2021/05/06
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.d.210422.001How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Shexiang Hai
PY  - 2021
DA  - 2021/05/06
TI  - The Arithmetic Operator of Fuzzy Regular Prismoid Numbers and Its Application to Fuzzy Risk Analysis
JO  - International Journal of Computational Intelligence Systems
SP  - 1541
EP  - 1563
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210422.001
DO  - 10.2991/ijcis.d.210422.001
ID  - Hai2021
ER  -