∎

11institutetext: Tapas Mondal 22institutetext: School of Basic Sciences,
Indian Institute of Technology Bhubaneswar
Tel.: +918345939869
22email<EMAIL_ADDRESS>33institutetext: Akshay Kumar Ojha 44institutetext:
School of Basic Sciences, Indian Institute of Technology Bhubaneswar
44email<EMAIL_ADDRESS>55institutetext: Sabyasachi Pani 66institutetext:
School of Basic Sciences, Indian Institute of Technology Bhubaneswar
66email<EMAIL_ADDRESS>
# Geometric Programming Problems with Triangular and Trapezoidal Two-fold
Uncertainty Distributions

Tapas Mondal  Akshay Kumar Ojha  Sabyasachi Pani  Tapas Mondal  Akshay Kumar
Ojha  Sabyasachi Pani

(Received: date / Accepted: date)

###### Abstract

Geometric programming (GP) is a well-known optimization tool for dealing with
a wide range of nonlinear optimization and engineering problems. In general,
it is assumed that the parameters of a GP problem are deterministic and
accurate. However, in the real-world GP problem, the parameters are frequently
inaccurate and ambiguous. This paper investigates the GP problem in an
uncertain environment, with the coefficients as triangular and trapezoidal
two-fold uncertain variables. In this paper, we introduce uncertain measures
in a generalized version and focus on more complicated two-fold uncertainties
to propose triangular and trapezoidal two-fold uncertain variables within the
context of uncertainty theory. We develop three reduction methods to convert
triangular and trapezoidal two-fold uncertain variables into single-fold
uncertain variables using optimistic, pessimistic, and expected value
criteria. Reduction methods are used to convert the GP problem with two-fold
uncertainty into the GP problem with single-fold uncertainty. Furthermore, the
chance-constrained uncertain-based framework is used to solve the reduced
single-fold uncertain GP problem. Finally, a numerical example is provided to
demonstrate the effectiveness of the procedures.

###### Keywords:

Geometric programming Triangular two-fold uncertain variable Trapezoidal two-
fold uncertain variable Reduction method

###### MSC:

90C30 90C46 90C47 49K45

##  1 Introduction

Geometric programming (GP) is one of the most effective methods for solving
nonlinear optimization problems. In 1967, Duffin et al. Duffin 1967  first
introduced the fundamental theories of the GP problem. Generally, the GP
technique is used for posynomial types of objective and constraint functions.
The GP problem with all positive parameters except the exponents is called the
posynomial problem. The GP technique converts the primal problem into a dual
problem, which is easier to solve.

Nowadays, the GP problem is used in engineering design problems like circuit
design Chu 2001 ; Hershenson 2001 , inventory modeling Mandal 2006 ; Roy 1997
; Worrall 1982 , production planning Cheng 1991 ; Choi 1996 ; Islam 2007 ;
Jung 2001 ; Kim 1998 ; Lee 1993 , risk management Scott 1995 , chemical
processing Passy 1968 ; Ruckaert 1978 ; Wall 1986 , information theory Chiang
2005 , and structural design Gupta 1986 .

The traditional GP problem assumes that the parameters are precisely known and
deterministic. Several researchers developed efficient and practical
algorithms for the conventional GP problem with specific and precise
parameters Avriel 1975 ; Beightler 1976 ; Duffin 1967 ; Duffin 1973 ; Fang
1988 ; Kortanek 1992 ; Kortanek 1997 ; Maranas 1997 ; Rajgopal 1990 ; Rajgopal
1992 ; Rajgopal 2002 . In practice, however, the values of the parameters of a
GP problem may be uncertain and imprecise. As a result, numerous methods to
the problem of uncertain GP have been developed. Avriel and Wilde Avriel 1969
introduced the stochastic GP problem. The authors considered the GP problem,
in which the coefficients of objective and constraint functions are
nonnegative random variables. Liu Liu 2006  devised a method to find the range
of the posynomial objective values for the GP problem with coefficients and
exponents that have interval values. Mahapatra and Mandal Mahapatra 2012
solved the GP problem with interval-valued coefficients in a parametric form.

In the last few decades, the GP problem has been developed in fuzzy
environments. In 1993, Cao Cao 1993  extended the GP problem in an imprecise
environment, along with interval and fuzzy coefficients. Later on, the same
author Cao 1997  proposed the GP problem with T-fuzzy coefficients. Mandal and
Roy Mandal 2006  solved the GP problem with L-R fuzzy coefficients. Yang and
Cao Yang 2007  made significant contributions to the fuzzy relational GP under
monomial. Liu Liu 2007  developed the GP problem with fuzzy parameters. Shiraz
et al. Shiraz 2017  used possibility, necessity, and credibility approaches to
solve the fuzzy chance-constrained GP problem. Under the rough set theory,
Shiraz and Fukuyama Shiraz 2018  developed the GP problem.

Recently, Liu Liu 2015  pioneered uncertainty theory, a new and developing
field of mathematics. In uncertainty theory, a measurable function from
uncertainty space to the set of real numbers is known as an uncertain variable
(UV). It is characterized in mathematics by its corresponding uncertainty
distribution (UD). UVs can be normal, linear, zigzag, log-normal, triangular,
or trapezoidal. Using uncertainty theory, many decision-making optimization
problems can be solved efficiently and effectively. To solve the GP problem,
several researchers developed an uncertainty-based framework based on
uncertainty theory. Shiraz et al. Shiraz 2016  first considered the GP problem
in an uncertain environment. Based on uncertainty theory, the authors
developed the deterministic form of an uncertain GP problem under normal,
linear, and zigzag uncertainties. In 2022, Mondal et al. Mondal 2022
developed a procedure for solving the GP problem with uncertainty. The authors
derived the equivalent deterministic form of the GP problem under triangular
and trapezoidal uncertainty.

It is noted that the GP problem is only taken into account when there is a
single-fold uncertainty. In practice, however, it is observed that uncertainty
frequently occurs in multiple aspects of real-world decision-making systems
due to a lack of prior information. Thus, two-fold uncertainty is developed in
order to describe multi-fold uncertainty. In multi-fold uncertainty, the main
aim is to reduce multi-fold UVs into single-fold UVs. In 2015, Yang et al.
Yang 2015  developed the reduction methods of normal, linear, zigzag, and log-
normal two-fold UVs. The authors applied those methods in a solid
transportation problem.

According to the literature review, there has been a lot of research on the
crisp and fuzzy GP problem. A GP problem with single-fold uncertainties has
also recently emerged. To the best of our knowledge, no previous work on the
GP problem with triangular and trapezoidal two-fold uncertainty has been done.
As a result, we attempt to investigate the GP problem with triangular and
trapezoidal two-fold uncertainty. The following are the main contributions we
made to this study.

I.

We propose triangular and trapezoidal two-fold UVs within the framework of
uncertainty theory by introducing uncertain measures in a generalized version
and focusing on more complicated two-fold uncertainties. Moreover, we consider
the GP problem in an uncertain environment, with the coefficients as
triangular and trapezoidal two-fold UVs.

II.

We develop three reduction methods to reduce triangular and trapezoidal two-
fold UVs into single-fold UVs with the help of optimistic, pessimistic, and
expected value criteria.

III.

Reduction methods are used to convert the GP problem with two-fold uncertainty
into the GP problem with single-fold uncertainty.

IV.

The chance-constrained uncertain-based framework is used to solve the reduced
single-fold uncertain GP problem.

V.

A numerical example is demonstrated to show the efficiency of the procedures.

The paper is organized as follows. Triangular and trapezoidal two-fold UVs are
proposed in Section 2. Reduction methods are developed in Section 3. The GP
problem with triangular and trapezoidal two-fold uncertain coefficients are
considered and developed the solving procedures in Section 4. In Section 5, a
numerical example is given. Finally, a conclusion on this work is incorporated
in Section 6.

##  2 Triangular and Trapezoidal Two-fold UVs

In this section, we propose triangular and trapezoidal two-fold UVs. To do so,
first, we discuss some preliminary concepts of single-fold uncertainty.

###  2.1 Single-fold uncertainty

Uncertainty theory is a new branch of mathematics that uses the concept of UVs
to describe the characteristics of single-fold uncertainties in order to model
degrees of belief. To define UVs, we must first understand the ideas of
uncertain measure and uncertainty space.

###### Definition 1

Liu 2015  Let $L$ be the $\sigma-$algebra over the nonempty set $\Gamma$. In
order to define UV, we need to first define the uncertain measure and
uncertainty space. The uncertain measure is the set function
$\mathcal{M}:L\rightarrow[0,1],$ which has the following axioms.

I.

(Normality Axiom) $\mathcal{M}\\{\Gamma\\}=1$.

II.

(Duality Axiom) $\mathcal{M}\\{A\\}+\mathcal{M}\\{A^{c}\\}=1$ for any event
$A\in L$.

III.

(Subadditivity Axiom)
$\mathcal{M}\big{\\{}\bigcup\limits_{i=1}^{\infty}A_{i}\big{\\}}\leq\sum\limits_{i=1}^{\infty}\mathcal{M}\\{A_{i}\\}$
for every countable sequence of events $\\{A_{i}\\}_{i=1}^{\infty}$.

Moreover, the triplet $(\Gamma,L,\mathcal{M})$ is known as the uncertainty
space.

###### Remark 1

Liu 2015  Uncertainty is defined as one’s personal level of belief that an
event will occur. It is determined by personal knowledge of the event. If the
state of knowledge changes, the uncertain measure will change.

The concept of UV is extremely useful for describing uncertainties. Here is
the definition of UV.

###### Definition 2

Liu 2015  A measurable function $\xi:(\Gamma,L,\mathcal{M})\to\mathbb{R}$ is
called a UV, where $(\Gamma,L,\mathcal{M})$ is the uncertainty space and
$\mathbb{R}$ is the set of real numbers, i.e., for any Borel set $B$,
$\\{\xi\in B\\}$ is an event.

Alternatively, a UV can be clearly characterized by its corresponding UD,
which is a carrier of uncertain information about the UV. In most cases,
knowing the UD rather than the UV itself is sufficient. Here is the
mathematical definition of UD.

###### Definition 3

Liu 2015  If $\Phi$ is the UD corresponding to a UV $\xi$, then $\Phi$ is
defined as

$\Phi(x)=\mathcal{M}\\{\xi\leq x\\},\forall x\in\mathbb{R}.$

Moreover, the necessary and sufficient condition for a UD function is stated
as follows.

###### Theorem 2.1

Liu 2015  A function $\Phi:\mathbb{R}\to[0,1]$ is a UD if and only if it is a
monotonic increasing function except $\Phi(x)=0$ and $\Phi(x)=1$.

It is simple to describe uncertain information geometrically using the UD in a
two-dimensional space. Several UVs, like normal, linear, zigzag, log-normal,
triangular, and trapezoidal, are used in real-world decision-making systems.
Linear UV has an important role in generalized uncertainty. Here is the
definition of linear UV.

###### Definition 4

Liu 2015  Let $\Phi$ be the UD corresponding to a UV $\xi$. The UV $\xi$ is
said to be linear if and only if $\Phi$ is defined as follows.

$\Phi(x)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\ \frac{x-a}{b-a},&a\leq
x\leq b;\\\ 1,&x\geq b;\\\ \end{array}\right.$

where $a,b\in\mathbb{R},a<b.$ If $\xi$ is a linear UV with the parameters $a$
and $b$, then we write it as $\xi\sim\mathcal{L}(a,b).$

In this research, our main work on triangular and trapezoidal uncertainty. In
2022, Mondal et al. Mondal 2022  proposed triangular and trapezoidal single-
fold UVs. Here are the definitions of those UVs.

###### Definition 5

Mondal 2022  Let $\Phi$ be the UD corresponding to a UV $\xi$. The UV $\xi$ is
said to be triangular if and only if $\Phi$ is defined as follows.

$\Phi(x)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\
\frac{(x-a)^{2}}{(b-a)(c-a)},&a\leq x\leq b;\\\
1-\frac{(c-x)^{2}}{(c-a)(c-b)},&b\leq x\leq c;\\\ 1,&x\geq c;\\\
\end{array}\right.$

where $a,b,c\in\mathbb{R},a<b<c.$ If $\xi$ is a triangular UV with the
parameters $a$, $b$, and $c$, then we write it as
$\xi\sim\mathcal{TRI}(a,b,c).$

###### Definition 6

Mondal 2022  Let $\Phi$ be the UD corresponding to a UV $\xi$. The UV $\xi$ is
said to be trapezoidal if and only if $\Phi$ is defined as follows.

$\Phi(x)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\
\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},&a\leq x\leq b;\\\
\frac{(2x-a-b)}{(d+c-a-b)},&b\leq x\leq c;\\\
1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},&c\leq x\leq d;\\\ 1,&x\geq d;\\\
\end{array}\right.$

where $a,b,c,d\in\mathbb{R},a<b<c<d.$ If $\xi$ is a trapezoidal UV with the
parameters $a$, $b$, $c$, and $d$, then we write it as
$\xi\sim\mathcal{TRA}(a,b,c,d).$

###  2.2 Two-fold uncertainty

In a real-world situation, there may be inconsistencies in various experts’
information on an event. Due to this lack of consistency, it is hard to give a
good estimate of how uncertain an event is. In this case, it makes sense to
generalize the uncertain measure by using UV values in the interval $[0,1]$
rather than constant values. The following discussion gives the concepts of
generalized uncertain measure, generalized uncertainty space, and two-fold
UVs. To begin, we must define a regular UV.

###### Definition 7

Liu 2015  Let $\Phi$ be the UD corresponding to a UV $\xi$. The UV $\xi$ is
said to be regular if and only if $\Phi$ is continuous and strictly increasing
with respect $x$ at which $0<\Phi(x)<1$, and

$\lim\limits_{x\to-\infty}\Phi(x)=0,\lim\limits_{x\to\infty}\Phi(x)=1.$

Alternatively, a UV $\xi$ is said to be regular if and only if the inverse
$\Phi^{-1}(\alpha)$ of the UD $\Phi(x)$ corresponding to the UV $\xi$ exists
and it is unique for each $\alpha\in(0,1)$.

Similar to Definition 1, we give the concept of generalized uncertain measure.

###### Definition 8

Yang 2015  Let $L$ be the $\sigma-$algebra over the nonempty set $\Gamma$. To
define two-fold UV, we need to first define the generalized uncertain measure
and generalized uncertainty space. The generalized uncertain measure is the
set function $\mathcal{\tilde{M}}:L\rightarrow[0,1],$ which has the following
two axioms.

Axiom 1.

$\mathcal{\tilde{M}}\\{\phi\\}=0,\mathcal{\tilde{M}}\\{\Gamma\\}=1$.

Axiom 2.

$\mathcal{\tilde{M}}\\{A\\}+\mathcal{\tilde{M}}\\{A^{c}\\}\leq 1$ for any
event $A\in L$.

Moreover, the triplet $(\Gamma,L,\mathcal{\tilde{M}})$ is known as the
generalized uncertainty space.

###### Remark 2

Yang 2015  The set function $\mathcal{\tilde{M}}$ is said to be a generalized
uncertain measure if $\mathcal{\tilde{M}}\\{A\\}$ is a regular UV for any
measurable set $A\in L$.

To describe two-fold uncertainties, the concept of two-fold UV is very useful.

###### Definition 9

Yang 2015  A two-fold UV $\tilde{\xi}$ is a measurable function from the
generalized uncertainty space $(\Gamma,L,{\mathcal{\tilde{M}}})$ to the set of
real numbers $\mathbb{R}$ such that $\\{\eta\in\Gamma|\tilde{\xi}(\eta)\in
B\\}\in L$ for any Borel set $B$.

A two-fold UV can also be defined by its UD function. Because we use regular
UVs to quantify the likelihood of uncertain events, the corresponding
distribution function is referred to as the two-fold UD function in the
discussion that follows.

###### Definition 10

Yang 2015  The mapping $\tilde{\Phi}_{\tilde{\xi}}:\mathbb{R}\times
S_{x}\to[0,1]$ is called a two-fold UD. If $\tilde{\Phi}_{\tilde{\xi}}$ is a
two-fold UD corresponding to a two-fold UV $\tilde{\xi}$, then
$\tilde{\Phi}_{\tilde{\xi}}$ is defined as

$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\mathcal{\tilde{M}}\\{\Phi_{\tilde{\xi}}(x)\leq
y\\},(x,y)\in\mathbb{R}\times S_{x},$

where $\Phi_{\tilde{\xi}}(x)=\mathcal{M}\\{\eta\in\Gamma|\tilde{\xi}(\eta)\leq
x\\}$, $S_{x}\subset[0,1]$ is the support of $\Phi_{\tilde{\xi}}(x)$, i.e.,

$S_{x}=\\{y\in[0,1]\big{|}0<\tilde{\Phi}_{\tilde{\xi}}(x,y)<1\\}.$

In this research, we propose triangular and trapezoidal two-fold UVs. The
proposed definitions are given as follows.

###### Definition 11

Let $\tilde{\xi}$ be a triangular two-fold UV, denoted by
$\mathcal{TRI}(a,b,c;\theta_{l},\theta_{r})$. The two-fold UD
$\tilde{\Phi}_{\tilde{\xi}}(x,y)$ of a triangular two-fold UV $\tilde{\xi}$ is
defined as follows.

I.

If $x\leq a$, then $\tilde{\Phi}_{\tilde{\xi}}(x,y)=0$.

II.

If $a<x<b$, then

$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\left\\{\begin{array}[]{ll}0,&y\leq A;\\\
\frac{y-A}{B-A},&A\leq y\leq B;\\\ 1,&y\geq B;\\\ \end{array}\right.$

where

$A=\frac{(x-a)^{2}}{(b-a)(c-a)}-\theta_{l}\min\bigg{\\{}\frac{(x-a)^{2}}{(b-a)(c-a)},\frac{b-a}{c-a}-\frac{(x-a)^{2}}{(b-a)(c-a)}\bigg{\\}},$
$B=\frac{(x-a)^{2}}{(b-a)(c-a)}+\theta_{r}\min\bigg{\\{}\frac{(x-a)^{2}}{(b-a)(c-a)},\frac{b-a}{c-a}-\frac{(x-a)^{2}}{(b-a)(c-a)}\bigg{\\}}.$

III.

If $x=b$, then $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\frac{b-a}{c-a}$.

IV.

If $b<x<c$, then

$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\left\\{\begin{array}[]{ll}0,&y\leq C;\\\
\frac{y-C}{D-C},&C\leq y\leq D;\\\ 1,&y\geq D;\\\ \end{array}\right.$

where

$C=1-\frac{(c-x)^{2}}{(c-a)(c-b)}-\theta_{l}\min\bigg{\\{}\frac{c-b}{c-a}-\frac{(c-x)^{2}}{(c-a)(c-b)},\frac{(c-x)^{2}}{(c-a)(c-b)}\bigg{\\}},$
$D=1-\frac{(c-x)^{2}}{(c-a)(c-b)}+\theta_{r}\min\bigg{\\{}\frac{c-b}{c-a}-\frac{(c-x)^{2}}{(c-a)(c-b)},\frac{(c-x)^{2}}{(c-a)(c-b)}\bigg{\\}}.$

V.

If $x\geq c$, then $\tilde{\Phi}_{\tilde{\xi}}(x,y)=1$.

In this definition, $\theta_{l},\theta_{r}\in[0,1]$ are two parameters that
characterize the degree of uncertainty of $\tilde{\xi}$ when $x$ is given.

###### Remark 3

Alternatively, the two-fold UD of a triangular two-fold UV $\tilde{\xi}$ can
be presented as

$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\
\mathcal{L}(A,B),&a<x<b;\\\ \frac{b-a}{c-a},&x=b;\\\
\mathcal{L}(C,D),&b<x<c;\\\ 1,&x\geq c;\\\ \end{array}\right.$

where $A,B,C,D$ are given in Definition 11.

###### Definition 12

Let $\tilde{\xi}$ be a trapezoidal two-fold UV, denoted by
$\mathcal{TRA}(a,b,c,d;\theta_{l},\theta_{r})$. The two-fold UD
$\tilde{\Phi}_{\tilde{\xi}}(x,y)$ of a trapezoidal two-fold UV $\tilde{\xi}$
is defined as follows.

I.

If $x\leq a$, then $\tilde{\Phi}_{\tilde{\xi}}(x,y)=0$.

II.

If $a<x<b$, then

$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\left\\{\begin{array}[]{ll}0,&y\leq A;\\\
\frac{y-A}{B-A},&A\leq y\leq B;\\\ 1,&y\geq B;\\\ \end{array}\right.$

where

$A=\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-\theta_{l}\min\bigg{\\{}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},\frac{b-a}{d+c-a-b}-\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}\bigg{\\}},$
$B=\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}+\theta_{r}\min\bigg{\\{}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},\frac{b-a}{d+c-a-b}-\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}\bigg{\\}}.$

III.

If $x=b$, then $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\frac{b-a}{d+c-a-b}$.

IV.

If $b<x<c$, then

$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\left\\{\begin{array}[]{ll}0,&y\leq C;\\\
\frac{y-C}{D-C},&C\leq y\leq D;\\\ 1,&y\geq D;\\\ \end{array}\right.$

where

$C=\frac{2x-a-b}{d+c-a-b}-\theta_{l}\min\bigg{\\{}\frac{2x-a-b}{d+c-a-b}-\frac{b-a}{d+c-a-b},\frac{2c-a-b}{d+c-a-b}-\frac{2x-a-b}{d+c-a-b}\bigg{\\}},$
$D=\frac{2x-a-b}{d+c-a-b}+\theta_{r}\min\bigg{\\{}\frac{2x-a-b}{d+c-a-b}-\frac{b-a}{d+c-a-b},\frac{2c-a-b}{d+c-a-b}-\frac{2x-a-b}{d+c-a-b}\bigg{\\}}.$

V.

If $x=c$, then $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\frac{2c-a-b}{d+c-a-b}$.

VI.

If $c<x<d$, then

$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\left\\{\begin{array}[]{ll}0,&y\leq E;\\\
\frac{y-E}{F-E},&E\leq y\leq F;\\\ 1,&y\geq F;\\\ \end{array}\right.$

where

$E=1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-\theta_{l}\min\bigg{\\{}\frac{d-c}{d+c-a-b}-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}\bigg{\\}},$
$F=1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}+\theta_{r}\min\bigg{\\{}\frac{d-c}{d+c-a-b}-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}\bigg{\\}}.$

VII.

If $x\geq d$, then $\tilde{\Phi}_{\tilde{\xi}}(x,y)=1$.

In this definition, $\theta_{l},\theta_{r}\in[0,1]$ are two parameters that
characterize the degree of uncertainty of $\tilde{\xi}$ when $x$ is given.

###### Remark 4

Similarly, the two-fold UD of a trapezoidal two-fold UV $\tilde{\xi}$ can be
presented as

$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\
\mathcal{L}(A,B),&a<x<b;\\\ \frac{b-a}{d+c-a-b},&x=b;\\\
\mathcal{L}(C,D),&b<x<c;\\\ \frac{2c-a-b}{d+c-a-b},&x=c;\\\
\mathcal{L}(E,F),&c<x<d;\\\ 1,&x\geq d;\\\ \end{array}\right.$

where $A,B,C,D,E,F$ are given in Definition 12.

##  3  Reduction Methods

In this section, we show how to reduce two-fold UV to single-fold UV. To
accomplish this, we employ three critical values criteria: optimistic,
pessimistic, and expected value criteria. To that end, we will first introduce
the concepts of critical values of single-fold UV. Furthermore, the critical
values of triangular and trapezoidal single-fold UVs are estimated.

###  3.1 Critical values of single-fold UVs

Liu Liu 2015  proposed the concepts critical values for single-fold UVs as
follows.

###### Definition 13

Liu 2015  If $\xi$ is a UV, then the optimistic, pessimistic, and expected
values of $\xi$ are defined as follows.

I.

$\xi_{\sup}(\alpha)=\sup\\{r|\mathcal{M}\\{\xi\geq
r\\}\geq\alpha\\},\alpha\in(0,1).$

II.

$\xi_{\inf}(\alpha)=\inf\\{r|\mathcal{M}\\{\xi\leq
r\\}\geq\alpha\\},\alpha\in(0,1).$

III.

$\xi_{exp}=\int\limits_{0}^{\infty}\mathcal{M}\\{\xi\geq
r\\}dr-\int\limits_{-\infty}^{0}\mathcal{M}\\{\xi\leq r\\}dr.$

Alternatively, Definition 13 can be represented in terms of UD. The following
theorem, known as the measure inversion theorem, is extremely useful in this
regard.

###### Theorem 3.1

Liu 2015  Let $\xi$ be a regular UV with UD $\Phi$. Then for any real number
$r$, $\mathcal{M}\\{\xi\leq r\\}=\Phi(r)\text{ and }\mathcal{M}\\{\xi\geq
r\\}=1-\Phi(r).$

Based on Theorem 3.1, Definition 13 can be redefined as

I.

$\xi_{\sup}(\alpha)=\sup\\{r|1-\Phi(r)\geq\alpha\\},\alpha\in(0,1).$

II.

$\xi_{\inf}(\alpha)=\inf\\{r|\Phi(r)\geq\alpha\\},\alpha\in(0,1).$

III.

$\xi_{exp}=\int\limits_{0}^{\infty}(1-\Phi(r))dr-\int\limits_{-\infty}^{0}\Phi(r)dr.$

###### Remark 5

Integrating by parts, we have
$\int\limits_{0}^{\infty}(1-\Phi(r))dr-\int\limits_{-\infty}^{0}\Phi(r)dr=\int\limits_{-\infty}^{\infty}r{\Phi}^{\prime}(r)dr.$
Therefore, the expected value of a UV $\xi$ can be determined by the formula
given as $\xi_{exp}=\int\limits_{-\infty}^{\infty}r{\Phi}^{\prime}(r)dr.$

Based on the above operations, the following useful theorem is developed.

###### Theorem 3.2

Liu 2015  Let $\xi\sim\mathcal{L}(a,b)$ be a linear UV. The optimistic,
pessimistic, and expected values of $\xi$ are as follows.

I.

$\xi_{\sup}(\alpha)=\alpha a+(1-\alpha)b,\alpha\in(0,1).$

II.

$\xi_{\inf}(\alpha)=(1-\alpha)a+\alpha b,\alpha\in(0,1).$

III.

$\xi_{exp}=\frac{a+b}{2}.$

Similarly, we developed the critical values of triangular and trapezoidal UVs.

###### Theorem 3.3

Let $\xi\sim\mathcal{TRI}(a,b,c)$ be a triangular UV. The optimistic,
pessimistic, and expected values of $\xi$ are as follows.

I.

$\xi_{\sup}(\alpha)=\left\\{\begin{array}[]{ll}a+\sqrt{(1-\alpha)(b-a)(c-a)},&0<\alpha\leq\frac{b-a}{c-a};\\\
c-\sqrt{\alpha(c-a)(c-b)},&\frac{b-a}{c-a}\leq\alpha<1.\\\ \end{array}\right.$

II.

$\xi_{\inf}(\alpha)=\left\\{\begin{array}[]{ll}a+\sqrt{\alpha(b-a)(c-a)},&0<\alpha\leq\frac{b-a}{c-a};\\\
c-\sqrt{(1-\alpha)(c-a)(c-b)},&\frac{b-a}{c-a}\leq\alpha<1.\\\
\end{array}\right.$

III.

$\xi_{exp}=\frac{a+b+c}{3}.$

###### Proof

We prove case (I) and case (III). The proof of case (II) is similar to the
proof of case (I).
For any $0<\alpha\leq\frac{b-a}{c-a},$ we have

$\displaystyle\xi_{\sup}(\alpha)$
$\displaystyle=\sup\\{r|1-\Phi(r)\geq\alpha\\}$
$\displaystyle=\sup\bigg{\\{}r\bigg{|}1-\frac{(r-a)^{2}}{(b-a)(c-a)}\geq\alpha\bigg{\\}}$
$\displaystyle=\sup\bigg{\\{}r\bigg{|}\frac{(r-a)^{2}}{(b-a)(c-a)}\leq
1-\alpha\bigg{\\}}$
$\displaystyle=\sup\\{r|(r-a)^{2}\leq(1-\alpha)(b-a)(c-a)\\}$
$\displaystyle=\sup\bigg{\\{}r|-\sqrt{(1-\alpha)(b-a)(c-a)}\leq(r-a)\leq\sqrt{(1-\alpha)(b-a)(c-a)}\bigg{\\}}$
$\displaystyle=\sup\bigg{\\{}r|a-\sqrt{(1-\alpha)(b-a)(c-a)}\leq r\leq
a+\sqrt{(1-\alpha)(b-a)(c-a)}\bigg{\\}}$
$\displaystyle=a+\sqrt{(1-\alpha)(b-a)(c-a)}.$

For any $\frac{b-a}{c-a}\leq\alpha<1,$ we have

$\displaystyle\xi_{\sup}(\alpha)$
$\displaystyle=\sup\\{r|1-\Phi(r)\geq\alpha\\}$
$\displaystyle=\sup\bigg{\\{}r\bigg{|}1-\bigg{[}1-\frac{(c-r)^{2}}{(c-a)(c-b)}\bigg{]}\geq\alpha\bigg{\\}}$
$\displaystyle=\sup\bigg{\\{}r\bigg{|}\frac{(c-r)^{2}}{(c-a)(c-b)}\geq\alpha\bigg{\\}}$
$\displaystyle=\sup\\{r|(c-r)^{2}\geq\alpha(c-a)(c-b)\\}$

If $c-r\leq-\sqrt{\alpha(c-a)(c-b)},$ then $r\geq c+\sqrt{\alpha(c-a)(c-b)}.$
So, $\Phi(r)=1,$ which implies $1-\Phi(r)=0\geq\alpha.$ This contradicts
$\alpha\in(0,1).$ Thus, we omit the case $c-r\leq-\sqrt{\alpha(c-a)(c-b)}.$
Therefore, we have

$\displaystyle\xi_{\sup}(\alpha)$
$\displaystyle=\sup\bigg{\\{}r|c-r\geq\sqrt{\alpha(c-a)(c-b)}\bigg{\\}}$
$\displaystyle=\sup\bigg{\\{}r|r\leq c-\sqrt{\alpha(c-a)(c-b)}\bigg{\\}}$
$\displaystyle=c-\sqrt{\alpha(c-a)(c-b)}.$

Hence the proof of case (I) is done.

Next, we proof the case (III). We have

$\displaystyle\xi_{exp}$
$\displaystyle=\int\limits_{-\infty}^{\infty}r{\Phi}^{\prime}(r)dr$
$\displaystyle=\int\limits_{a}^{b}r\cdot\frac{2(r-a)}{(b-a)(c-a)}dr+\int\limits_{b}^{c}r\cdot\frac{2(c-r)}{(c-a)(c-b)}dr$
$\displaystyle=\frac{2b^{2}-a^{2}-ab}{3(c-a)}+\frac{c^{2}+bc-2b^{2}}{3(c-a)}$
$\displaystyle=\frac{a+b+c}{3}.$

Hence the proof of case (III) is done.

###### Theorem 3.4

Let $\xi\sim\mathcal{TRA}(a,b,c,d)$ be a trapezoidal UV. The optimistic,
pessimistic, and expected values of $\xi$ are as follows.

I.

$\xi_{\sup}(\alpha)=\left\\{\begin{array}[]{ll}a+\sqrt{(1-\alpha)(d+c-a-b)(b-a)},&0<\alpha\leq\frac{b-a}{d+c-a-b};\\\
\frac{(a+b)+(1-\alpha)(d+c-a-b)}{2},&\frac{b-a}{d+c-a-b}\leq\alpha\leq\frac{2c-a-b}{d+c-a-b};\\\
d-\sqrt{\alpha(d+c-a-b)(d-c)},&\frac{2c-a-b}{d+c-a-b}\leq\alpha<1.\\\
\end{array}\right.$

II.

$\xi_{\inf}(\alpha)=\left\\{\begin{array}[]{ll}a+\sqrt{\alpha(d+c-a-b)(b-a)},&0<\alpha\leq\frac{b-a}{d+c-a-b};\\\
\frac{(a+b)+\alpha(d+c-a-b)}{2},&\frac{b-a}{d+c-a-b}\leq\alpha\leq\frac{2c-a-b}{d+c-a-b};\\\
d-\sqrt{(1-\alpha)(d+c-a-b)(d-c)},&\frac{2c-a-b}{d+c-a-b}\leq\alpha<1.\\\
\end{array}\right.$

III.

$\xi_{exp}=\frac{1}{3(d+c-a-b)}\bigg{[}\frac{d^{3}-c^{3}}{d-c}-\frac{b^{3}-a^{3}}{b-a}\bigg{]}.$

###### Proof

The proof is similar to the proof of Theorem 3.3.

###  3.2 Reduction methods of two-fold UVs with critical values criteria

Using optimistic, pessimistic, and expected value criteria, we develop three
different reduction methods to convert triangular two-fold UV into single-fold
UV. The following theorems are developed in this regard.

###### Theorem 3.5

Let $\tilde{\xi}\sim\mathcal{TRI}(a,b,c;\theta_{l},\theta_{r})$ be a
triangular two-fold UV. The reduced single-fold UD via $\alpha$ optimistic
value criteria for any $\alpha\in(0,1)$ is as follows.

I.

If $x\leq a$, then $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=0.$

II.

If $a<x<b$, then

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\left\\{\begin{array}[]{ll}\frac{(x-a)^{2}}{(b-a)(c-a)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\frac{(x-a)^{2}}{(b-a)(c-a)},&a<x\leq
a+\frac{b-a}{\sqrt{2}};\\\
\frac{(x-a)^{2}}{(b-a)(c-a)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{b-a}{c-a}-\frac{(x-a)^{2}}{(b-a)(c-a)}\big{]},&a+\frac{b-a}{\sqrt{2}}\leq
x<b.\\\ \end{array}\right.$

III.

If $x=b$, then $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\frac{b-a}{c-a}.$

IV.

If $b<x<c$, then

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\left\\{\begin{array}[]{ll}1-\frac{(c-x)^{2}}{(c-a)(c-b)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{c-b}{c-a}-\frac{(c-x)^{2}}{(c-a)(c-b)}\big{]},&b<x\leq
c-\frac{c-b}{\sqrt{2}};\\\
1-\frac{(c-x)^{2}}{(c-a)(c-b)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\frac{(c-x)^{2}}{(c-a)(c-b)},&c-\frac{c-b}{\sqrt{2}}\leq
x<c.\\\ \end{array}\right.$

V.

If $x\geq c$, then $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=1.$

###### Proof

Let $\alpha\in(0,1)$.

If $x\leq a,$ then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=0.$ Therefore,
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=0,\forall x\leq a.$

If $a<x<b$, then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\mathcal{L}(A,B),$
where $A,B$ are given in Definition 11. So, based on Theorem 3.2, we have

$\displaystyle\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)$ $\displaystyle=\alpha
A+(1-\alpha)B$
$\displaystyle=\alpha\bigg{[}\frac{(x-a)^{2}}{(b-a)(c-a)}-\theta_{l}\min\bigg{\\{}\frac{(x-a)^{2}}{(b-a)(c-a)},\frac{b-a}{c-a}-\frac{(x-a)^{2}}{(b-a)(c-a)}\bigg{\\}}\bigg{]}$
$\displaystyle\qquad+(1-\alpha)\bigg{[}\frac{(x-a)^{2}}{(b-a)(c-a)}+\theta_{r}\min\bigg{\\{}\frac{(x-a)^{2}}{(b-a)(c-a)},\frac{b-a}{c-a}-\frac{(x-a)^{2}}{(b-a)(c-a)}\bigg{\\}}\bigg{]}$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{(x-a)^{2}}{(b-a)(c-a)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\frac{(x-a)^{2}}{(b-a)(c-a)},&a<x\leq
a+\frac{b-a}{\sqrt{2}};\\\
\frac{(x-a)^{2}}{(b-a)(c-a)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{b-a}{c-a}-\frac{(x-a)^{2}}{(b-a)(c-a)}\big{]},&a+\frac{b-a}{\sqrt{2}}\leq
x<b.\\\ \end{array}\right.$

If $x=b,$ then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\frac{b-a}{c-a}.$
Therefore, $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\frac{b-a}{c-a}$ for $x=b.$

If $b<x<c$, then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\mathcal{L}(C,D),$
where $C,D$ are given in Definition 11. So, based on Theorem 3.2, we have

$\displaystyle\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)$ $\displaystyle=\alpha
C+(1-\alpha)D$
$\displaystyle=\alpha\bigg{[}1-\frac{(c-x)^{2}}{(c-a)(c-b)}-\theta_{l}\min\bigg{\\{}\frac{c-b}{c-a}-\frac{(c-x)^{2}}{(c-a)(c-b)},\frac{(c-x)^{2}}{(c-a)(c-b)}\bigg{\\}}\bigg{]}$
$\displaystyle\qquad+(1-\alpha)\bigg{[}1-\frac{(c-x)^{2}}{(c-a)(c-b)}+\theta_{r}\min\bigg{\\{}\frac{c-b}{c-a}-\frac{(c-x)^{2}}{(c-a)(c-b)},\frac{(c-x)^{2}}{(c-a)(c-b)}\bigg{\\}}\bigg{]}$
$\displaystyle=\left\\{\begin{array}[]{ll}1-\frac{(c-x)^{2}}{(c-a)(c-b)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{c-b}{c-a}-\frac{(c-x)^{2}}{(c-a)(c-b)}\big{]},&b<x\leq
c-\frac{c-b}{\sqrt{2}};\\\
1-\frac{(c-x)^{2}}{(c-a)(c-b)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\frac{(c-x)^{2}}{(c-a)(c-b)},&c-\frac{c-b}{\sqrt{2}}\leq
x<c.\\\ \end{array}\right.$

If $x\geq c,$ then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=1.$ Therefore,
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=1,\forall x\geq c.$

###### Theorem 3.6

Let $\tilde{\xi}\sim\mathcal{TRI}(a,b,c;\theta_{l},\theta_{r})$ be a
triangular two-fold UV. The reduced single-fold UD via $\alpha$ pessimistic
value criteria for any $\alpha\in(0,1)$ is as follows.

I.

If $x\leq a$, then $\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=0.$

II.

If $a<x<b$, then

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}\frac{(x-a)^{2}}{(b-a)(c-a)}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\frac{(x-a)^{2}}{(b-a)(c-a)},&a<x\leq
a+\frac{b-a}{\sqrt{2}};\\\
\frac{(x-a)^{2}}{(b-a)(c-a)}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\big{[}\frac{b-a}{c-a}-\frac{(x-a)^{2}}{(b-a)(c-a)}\big{]},&a+\frac{b-a}{\sqrt{2}}\leq
x<b.\\\ \end{array}\right.$

III.

If $x=b$, then $\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\frac{b-a}{c-a}.$

IV.

If $b<x<c$, then

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}1-\frac{(c-x)^{2}}{(c-a)(c-b)}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\big{[}\frac{c-b}{c-a}-\frac{(c-x)^{2}}{(c-a)(c-b)}\big{]},&b<x\leq
c-\frac{c-b}{\sqrt{2}};\\\
1-\frac{(c-x)^{2}}{(c-a)(c-b)}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\frac{(c-x)^{2}}{(c-a)(c-b)},&c-\frac{c-b}{\sqrt{2}}\leq
x<c.\\\ \end{array}\right.$

V.

If $x\geq c$, then $\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=1.$

###### Proof

Based on Theorem 3.2, we have

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\
(1-\alpha)A+\alpha B,&a<x<b;\\\ \frac{b-a}{c-a},&x=b;\\\ (1-\alpha)C+\alpha
D,&b<x<c;\\\ 1,&x\geq c;\\\ \end{array}\right.\text{ for any }\alpha\in(0,1).$

Therefore, we get
$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\Phi_{\tilde{\xi}_{\sup}}(x;1-\alpha)$.
Using this result, the cases from (I) to (V) can be easily proved from Theorem
3.5.

###### Theorem 3.7

Let $\tilde{\xi}\sim\mathcal{TRI}(a,b,c;\theta_{l},\theta_{r})$ be a
triangular two-fold UV. The reduced single-fold UD via expected value criteria
is as follows.

I.

If $x\leq a$, then $\Phi_{\tilde{\xi}_{exp}}(x)=0.$

II.

If $a<x<b$, then

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}\frac{(x-a)^{2}}{(b-a)(c-a)}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\frac{(x-a)^{2}}{(b-a)(c-a)},&a<x\leq
a+\frac{b-a}{\sqrt{2}};\\\
\frac{(x-a)^{2}}{(b-a)(c-a)}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\big{[}\frac{b-a}{c-a}-\frac{(x-a)^{2}}{(b-a)(c-a)}\big{]},&a+\frac{b-a}{\sqrt{2}}\leq
x<b.\\\ \end{array}\right.$

III.

If $x=b$, then $\Phi_{\tilde{\xi}_{exp}}(x)=\frac{b-a}{c-a}.$

IV.

If $b<x<c$, then

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}1-\frac{(c-x)^{2}}{(c-a)(c-b)}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\big{[}\frac{c-b}{c-a}-\frac{(c-x)^{2}}{(c-a)(c-b)}\big{]},&b<x\leq
c-\frac{c-b}{\sqrt{2}};\\\
1-\frac{(c-x)^{2}}{(c-a)(c-b)}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\frac{(c-x)^{2}}{(c-a)(c-b)},&c-\frac{c-b}{\sqrt{2}}\leq
x<c.\\\ \end{array}\right.$

V.

If $x\geq c$, then $\Phi_{\tilde{\xi}_{exp}}(x)=1.$

###### Proof

Based on Theorem 3.2, we have

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\
\frac{A+B}{2},&a<x<b;\\\ \frac{b-a}{c-a},&x=b;\\\ \frac{C+D}{2},&b<x<c;\\\
1,&x\geq c.\\\ \end{array}\right.$

Therefore, we get
$\Phi_{\tilde{\xi}_{exp}}(x)=\Phi_{\tilde{\xi}_{\inf}}(x;\frac{1}{2})=\Phi_{\tilde{\xi}_{\sup}}(x;\frac{1}{2})$.
With this result, it is easy to get the cases from (I) to (V) from Theorem 3.5
and Theorem 3.6.

###### Remark 6

For any $\alpha\in(0,1),$ $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)$ is monotonic
increasing function except $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=0,\forall
x\leq a$ and $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=1,\forall x\geq c.$
According to Theorem 2.1, $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)$ is a single-
fold UD. Similarly,
$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha),\Phi_{\tilde{\xi}_{\exp}}(x)$ are also
single-fold UDs. As a result, the reduction strategies we provide are
acceptable.

To understand the efficiency of the reduction methods, we provide an example
of a triangular two-fold UV below.

###### Example 1

If $\tilde{\xi}=\mathcal{TRI}(2,4,5;0.5,0.6)$ is a triangular two-fold UV,
then

I.

the reduced single-fold UD via $\alpha$ optimistic value criteria is

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\left\\{\begin{array}[]{ll}0,&x\leq 2;\\\
\frac{(x-2)^{2}}{6}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\frac{(x-2)^{2}}{6},&2<x\leq
2+\sqrt{2};\\\
\frac{(x-2)^{2}}{6}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\big{[}\frac{2}{3}-\frac{(x-2)^{2}}{6}\big{]},&2+\sqrt{2}\leq
x<4;\\\ \frac{2}{3},&x=4;\\\
1-\frac{(5-x)^{2}}{3}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\big{[}\frac{1}{3}-\frac{(5-x)^{2}}{3}\big{]},&4<x\leq
5-\frac{1}{\sqrt{2}};\\\
1-\frac{(5-x)^{2}}{3}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\frac{(5-x)^{2}}{3},&5-\frac{1}{\sqrt{2}}\leq
x<5;\\\ 1,&x\geq 5;\\\ \end{array}\right.\text{ for any }\alpha\in(0,1);$

II.

the reduced single-fold UD via $\alpha$ pessimistic value criteria is

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}0,&x\leq 2;\\\
\frac{(x-2)^{2}}{6}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\frac{(x-2)^{2}}{6},&2<x\leq
2+\sqrt{2};\\\
\frac{(x-2)^{2}}{6}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\big{[}\frac{2}{3}-\frac{(x-2)^{2}}{6}\big{]},&2+\sqrt{2}\leq
x<4;\\\ \frac{2}{3},&x=4;\\\
1-\frac{(5-x)^{2}}{3}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\big{[}\frac{1}{3}-\frac{(5-x)^{2}}{3}\big{]},&4<x\leq
5-\frac{1}{\sqrt{2}};\\\
1-\frac{(5-x)^{2}}{3}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\frac{(5-x)^{2}}{3},&5-\frac{1}{\sqrt{2}}\leq
x<5;\\\ 1,&x\geq 5;\\\ \end{array}\right.\text{ for any }\alpha\in(0,1);$

III.

the reduced single-fold UD via expected value criteria is

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 2;\\\
\frac{(x-2)^{2}}{6}+\frac{1}{20}\frac{(x-2)^{2}}{6},&2<x\leq 2+\sqrt{2};\\\
\frac{(x-2)^{2}}{6}+\frac{1}{20}\big{[}\frac{2}{3}-\frac{(x-2)^{2}}{6}\big{]},&2+\sqrt{2}\leq
x<4;\\\ \frac{2}{3},&x=4;\\\
1-\frac{(5-x)^{2}}{3}+\frac{1}{20}\big{[}\frac{1}{3}-\frac{(5-x)^{2}}{3}\big{]},&4<x\leq
5-\frac{1}{\sqrt{2}};\\\
1-\frac{(5-x)^{2}}{3}+\frac{1}{20}\frac{(5-x)^{2}}{3},&5-\frac{1}{\sqrt{2}}\leq
x<5;\\\ 1,&x\geq 5.\\\ \end{array}\right.$

Here, Figures 1,2, and 3 show the reduced single-fold UD of the triangular
two-fold UV $\tilde{\xi}=\mathcal{TRI}(2,4,5;0.5,0.6)$ using optimistic,
pessimistic, and expected value criteria. Figures 1 and 2 are given as solid
figures because they are essentially bi-variate functions in terms of $x$ and
$\alpha$. For different values of $\alpha\in(0,1)$, the reduced UDs via
optimistic and pessimistic value criteria are shown in Figures 4 and 5 in 2D
pictures for clarity. It is observed that for each fixed $\alpha$, the related
curve satisfies the conditions in Theorem 2.1, corresponding to a UD.

Figure 1: Reduced single-fold UD of a triangular two-fold UD via $\alpha$
optimistic value criteria. Figure 2: Reduced single-fold UD of a triangular
two-fold UD via $\alpha$ pessimistic value criteria. Figure 3: Reduced single-
fold UD of a triangular two-fold UD via expected value criteria. Figure 4:
Reduced single-fold UDs of a triangular two-fold UD via optimistic value
criteria for different values of $\alpha$. Figure 5: Reduced single-fold UDs
of a triangular two-fold UD via pessimistic value criteria for different
values of $\alpha$.

Similar to earlier, using optimistic, pessimistic, and expected value
criteria, we develop three different reduction methods to convert trapezoidal
two-fold UV into single-fold UV. The following theorems are developed in this
regard.

###### Theorem 3.8

Let $\tilde{\xi}\sim\mathcal{TRA}(a,b,c,d;\theta_{l},\theta_{r})$ be a
trapezoidal two-fold UV. The reduced single-fold UD via $\alpha$ optimistic
value criteria for any $\alpha\in(0,1)$ is as follows.

I.

If $x\leq a$, then $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=0.$

II.

If $a<x<b$, then

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\left\\{\begin{array}[]{ll}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},&a<x\leq
a+\frac{b-a}{\sqrt{2}};\\\
\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{b-a}{d+c-a-b}-\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}\big{]},&a+\frac{b-a}{\sqrt{2}}\leq
x<b.\\\ \end{array}\right.$

III.

If $x=b$, then $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\frac{b-a}{d+c-a-b}.$

IV.

If $b<x<c$, then

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\left\\{\begin{array}[]{ll}\frac{2x-a-b}{d+c-a-b}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{2x-a-b}{d+c-a-b}-\frac{b-a}{d+c-a-b}\big{]},&b<x\leq\frac{b+c}{2};\\\
\frac{2x-a-b}{d+c-a-b}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{2c-a-b}{d+c-a-b}-\frac{2x-a-b}{d+c-a-b}\big{]},&\frac{b+c}{2}\leq
x<c.\\\ \end{array}\right.$

V.

If $x=c$, then $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\frac{2c-a-b}{d+c-a-b}.$

VI.

If $c<x<d$, then

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\left\\{\begin{array}[]{ll}1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{d-c}{d+c-a-b}-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}\big{]},&c<x\leq
d-\frac{d-c}{\sqrt{2}};\\\
1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},&d-\frac{d-c}{\sqrt{2}}\leq
x<d.\\\ \end{array}\right.$

VII.

If $x\geq d$, then $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=1.$

###### Proof

Let $\alpha\in(0,1)$.

If $x\leq a,$ then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=0.$ Therefore,
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=0,\forall x\leq a.$

If $a<x<b$, then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\mathcal{L}(A,B),$
where $A,B$ are given in Definition 12. So, based on Theorem 3.2, we have
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\alpha A+(1-\alpha)B$

$\displaystyle=\alpha\bigg{[}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-\theta_{l}\min\bigg{\\{}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},\frac{b-a}{d+c-a-b}-\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}\bigg{\\}}\bigg{]}+$
$\displaystyle(1-\alpha)\bigg{[}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}+\theta_{r}\min\bigg{\\{}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},\frac{b-a}{d+c-a-b}-\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}\bigg{\\}}\bigg{]}$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},&a<x\leq
a+\frac{b-a}{\sqrt{2}};\\\
\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{b-a}{d+c-a-b}-\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}\big{]},&a+\frac{b-a}{\sqrt{2}}\leq
x<b.\\\ \end{array}\right.$

If $x=b,$ then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\frac{b-a}{d+c-a-b}.$
Therefore, $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\frac{b-a}{d+c-a-b}$ for
$x=b.$

If $b<x<c$, then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\mathcal{L}(C,D),$
where $C,D$ are given in Definition 12. So, based on Theorem 3.2, we have
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\alpha C+(1-\alpha)D$

$\displaystyle=\alpha\bigg{[}\frac{2x-a-b}{d+c-a-b}-\theta_{l}\min\bigg{\\{}\frac{2x-a-b}{d+c-a-b}-\frac{b-a}{d+c-a-b},\frac{2c-a-b}{d+c-a-b}-\frac{2x-a-b}{d+c-a-b}\bigg{\\}}\bigg{]}+$
$\displaystyle(1-\alpha)\bigg{[}\frac{2x-a-b}{d+c-a-b}+\theta_{r}\min\bigg{\\{}\frac{2x-a-b}{d+c-a-b}-\frac{b-a}{d+c-a-b},\frac{2c-a-b}{d+c-a-b}-\frac{2x-a-b}{d+c-a-b}\bigg{\\}}\bigg{]}$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{2x-a-b}{d+c-a-b}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{2x-a-b}{d+c-a-b}-\frac{b-a}{d+c-a-b}\big{]},&b<x\leq\frac{b+c}{2};\\\
\frac{2x-a-b}{d+c-a-b}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{2c-a-b}{d+c-a-b}-\frac{2x-a-b}{d+c-a-b}\big{]},&\frac{b+c}{2}\leq
x<c.\\\ \end{array}\right.$

If $x=c,$ then we have
$\tilde{\Phi}_{\tilde{\xi}}(x,y)=\frac{2c-a-b}{d+c-a-b}.$ Therefore,
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\frac{2c-a-b}{d+c-a-b}$ for $x=c.$

If $c<x<d$, then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=\mathcal{L}(E,F),$
where $E,F$ are given in Definition 12. So, based on Theorem 3.2, we have
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\alpha E+(1-\alpha)F$

$\displaystyle=\alpha\bigg{[}1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-\theta_{l}\min\bigg{\\{}\frac{d-c}{d+c-a-b}-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}\bigg{\\}}\bigg{]}+$
$\displaystyle(1-\alpha)\bigg{[}1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}+\theta_{r}\min\bigg{\\{}\frac{d-c}{d+c-a-b}-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}\bigg{\\}}\bigg{]}$
$\displaystyle=\left\\{\begin{array}[]{ll}1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\big{[}\frac{d-c}{d+c-a-b}-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}\big{]},&c<x\leq
d-\frac{d-c}{\sqrt{2}};\\\
1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-[\alpha\theta_{l}-(1-\alpha)\theta_{r}]\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},&d-\frac{d-c}{\sqrt{2}}\leq
x<d.\\\ \end{array}\right.$

If $x\geq d,$ then we have $\tilde{\Phi}_{\tilde{\xi}}(x,y)=1.$ Therefore,
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=1,\forall x\geq d.$

###### Theorem 3.9

Let $\tilde{\xi}\sim\mathcal{TRA}(a,b,c,d;\theta_{l},\theta_{r})$ be a
trapezoidal two-fold UV. The reduced single-fold UD via $\alpha$ pessimistic
value criteria for any $\alpha\in(0,1)$ is as follows.

I.

If $x\leq a$, then $\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=0.$

II.

If $a<x<b$, then

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},&a<x\leq
a+\frac{b-a}{\sqrt{2}};\\\
\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\big{[}\frac{b-a}{d+c-a-b}-\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}\big{]},&a+\frac{b-a}{\sqrt{2}}\leq
x<b.\\\ \end{array}\right.$

III.

If $x=b$, then $\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\frac{b-a}{d+c-a-b}.$

IV.

If $b<x<c$, then

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}\frac{2x-a-b}{d+c-a-b}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\big{[}\frac{2x-a-b}{d+c-a-b}-\frac{b-a}{d+c-a-b}\big{]},&b<x\leq\frac{b+c}{2};\\\
\frac{2x-a-b}{d+c-a-b}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\big{[}\frac{2c-a-b}{d+c-a-b}-\frac{2x-a-b}{d+c-a-b}\big{]},&\frac{b+c}{2}\leq
x<c.\\\ \end{array}\right.$

V.

If $x=c$, then $\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\frac{2c-a-b}{d+c-a-b}.$

VI.

If $c<x<d$, then

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\big{[}\frac{d-c}{d+c-a-b}-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}\big{]},&c<x\leq
d-\frac{d-c}{\sqrt{2}};\\\
1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-[(1-\alpha)\theta_{l}-\alpha\theta_{r}]\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},&d-\frac{d-c}{\sqrt{2}}\leq
x<d.\\\ \end{array}\right.$

VII.

If $x\geq d$, then $\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=1.$

###### Proof

Based on Theorem 3.2, we have

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\
(1-\alpha)A+\alpha B,&a<x<b;\\\ \frac{b-a}{d+c-a-b},&x=b;\\\
(1-\alpha)C+\alpha D,&b<x<c;\\\ \frac{2c-a-b}{d+c-a-b},&x=c;\\\
(1-\alpha)E+\alpha F,&c<x<d;\\\ 1,&x\geq d;\\\ \end{array}\right.\text{ for
any }\alpha\in(0,1).$

Therefore, we get
$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\Phi_{\tilde{\xi}_{\sup}}(x;1-\alpha)$.
Using this result, the cases from (I) to (VII) can be easily proved from
Theorem 3.8.

###### Theorem 3.10

Let $\tilde{\xi}\sim\mathcal{TRA}(a,b,c,d;\theta_{l},\theta_{r})$ be a
trapezoidal two-fold UV. The reduced single-fold UD via expected value
criteria is as follows.

I.

If $x\leq a$, then $\Phi_{\tilde{\xi}_{exp}}(x)=0.$

II.

If $a<x<b$, then

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\frac{(x-a)^{2}}{(d+c-a-b)(b-a)},&a<x\leq
a+\frac{b-a}{\sqrt{2}};\\\
\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\big{[}\frac{b-a}{d+c-a-b}-\frac{(x-a)^{2}}{(d+c-a-b)(b-a)}\big{]},&a+\frac{b-a}{\sqrt{2}}\leq
x<b.\\\ \end{array}\right.$

III.

If $x=b$, then $\Phi_{\tilde{\xi}_{exp}}(x)=\frac{b-a}{d+c-a-b}.$

IV.

If $b<x<c$, then

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}\frac{2x-a-b}{d+c-a-b}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\big{[}\frac{2x-a-b}{d+c-a-b}-\frac{b-a}{d+c-a-b}\big{]},&b<x\leq\frac{b+c}{2};\\\
\frac{2x-a-b}{d+c-a-b}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\big{[}\frac{2c-a-b}{d+c-a-b}-\frac{2x-a-b}{d+c-a-b}\big{]},&\frac{b+c}{2}\leq
x<c.\\\ \end{array}\right.$

V.

If $x=c$, then $\Phi_{\tilde{\xi}_{exp}}(x)=\frac{2c-a-b}{d+c-a-b}.$

VI.

If $c<x<d$, then

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\big{[}\frac{d-c}{d+c-a-b}-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}\big{]},&c<x\leq
d-\frac{d-c}{\sqrt{2}};\\\
1-\frac{(d-x)^{2}}{(d+c-a-b)(d-c)}-\frac{(\theta_{l}-\theta_{r})}{2}\cdot\frac{(d-x)^{2}}{(d+c-a-b)(d-c)},&d-\frac{d-c}{\sqrt{2}}\leq
x<d.\\\ \end{array}\right.$

VII.

If $x\geq d$, then $\Phi_{\tilde{\xi}_{exp}}(x)=1.$

###### Proof

Based on Theorem 3.2, we have

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq a;\\\
\frac{A+B}{2},&a<x<b;\\\ \frac{b-a}{d+c-a-b},&x=b;\\\ \frac{C+D}{2},&b<x<c;\\\
\frac{2c-a-b}{d+c-a-b},&x=c;\\\ \frac{E+F}{2},&c<x<d;\\\ 1,&x\geq d.\\\
\end{array}\right.$

Therefore, we get
$\Phi_{\tilde{\xi}_{exp}}(x)=\Phi_{\tilde{\xi}_{\inf}}(x;\frac{1}{2})=\Phi_{\tilde{\xi}_{\sup}}(x;\frac{1}{2})$.
With this result, it is easy to get the cases from (I) to (VII) from Theorem
3.8 and Theorem 3.9.

###### Remark 7

Similar to earlier remark, for any $\alpha\in(0,1),$
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)$ is monotonic increasing function except
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=0,\forall x\leq a$ and
$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=1,\forall x\geq d.$ According to Theorem
2.1, $\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)$ is a single-fold UD. Similarly,
$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha),\Phi_{\tilde{\xi}_{\exp}}(x)$ are also
single-fold UDs. As a result, our recommended strategies for reduction are
acceptable.

Here also, we provide an example of a trapezoidal two-fold UV to understand
the efficiency of the reduction methods.

###### Example 2

If $\tilde{\xi}=\mathcal{TRA}(2,4,6,8;0.5,0.6)$ is a trapezoidal two-fold UV,
then

I.

the reduced single-fold UD via $\alpha$ optimistic value criteria is

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\left\\{\begin{array}[]{ll}0,&x\leq 2;\\\
\frac{(x-2)^{2}}{16}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\frac{(x-2)^{2}}{16},&2<x\leq
2+\sqrt{2};\\\
\frac{(x-2)^{2}}{16}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\big{[}\frac{1}{4}-\frac{(x-2)^{2}}{16}\big{]},&2+\sqrt{2}\leq
x<4;\\\ \frac{1}{4},&x=4;\\\
\frac{2x-6}{8}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\big{[}\frac{2x-6}{8}-\frac{1}{4}\big{]},&4<x\leq
5;\\\
\frac{2x-6}{8}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\big{[}\frac{3}{4}-\frac{2x-6}{8}\big{]},&5\leq
x<6;\\\ \frac{3}{4},&x=6;\\\
1-\frac{(8-x)^{2}}{16}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\big{[}\frac{1}{4}-\frac{(8-x)^{2}}{16}\big{]},&6<x\leq
8-\sqrt{2};\\\
1-\frac{(8-x)^{2}}{16}-\big{(}\frac{11}{10}\alpha-\frac{3}{5}\big{)}\frac{(8-x)^{2}}{16},&8-\sqrt{2}\leq
x<8;\\\ 1,&x\geq 8;\\\ \end{array}\right.\text{ for any }\alpha\in(0,1);$

II.

the reduced single-fold UD via $\alpha$ pessimistic value criteria is

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\left\\{\begin{array}[]{ll}0,&x\leq 2;\\\
\frac{(x-2)^{2}}{16}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\frac{(x-2)^{2}}{16},&2<x\leq
2+\sqrt{2};\\\
\frac{(x-2)^{2}}{16}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\big{[}\frac{1}{4}-\frac{(x-2)^{2}}{16}\big{]},&2+\sqrt{2}\leq
x<4;\\\ \frac{1}{4},&x=4;\\\
\frac{2x-6}{8}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\big{[}\frac{2x-6}{8}-\frac{1}{4}\big{]},&4<x\leq
5;\\\
\frac{2x-6}{8}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\big{[}\frac{3}{4}-\frac{2x-6}{8}\big{]},&5\leq
x<6;\\\ \frac{3}{4},&x=6;\\\
1-\frac{(8-x)^{2}}{16}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\big{[}\frac{1}{4}-\frac{(8-x)^{2}}{16}\big{]},&6<x\leq
8-\sqrt{2};\\\
1-\frac{(8-x)^{2}}{16}-\big{(}\frac{1}{2}-\frac{11}{10}\alpha\big{)}\frac{(8-x)^{2}}{16},&8-\sqrt{2}\leq
x<8;\\\ 1,&x\geq 8;\\\ \end{array}\right.\text{ for any }\alpha\in(0,1);$

III.

the reduced single-fold UD via expected value criteria is

$\Phi_{\tilde{\xi}_{exp}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 2;\\\
\frac{(x-2)^{2}}{16}+\frac{1}{20}\frac{(x-2)^{2}}{16},&2<x\leq 2+\sqrt{2};\\\
\frac{(x-2)^{2}}{16}+\frac{1}{20}\big{[}\frac{1}{4}-\frac{(x-2)^{2}}{16}\big{]},&2+\sqrt{2}\leq
x<4;\\\ \frac{1}{4},&x=4;\\\
\frac{2x-6}{8}+\frac{1}{20}\big{[}\frac{2x-6}{8}-\frac{1}{4}\big{]},&4<x\leq
5;\\\
\frac{2x-6}{8}+\frac{1}{20}\big{[}\frac{3}{4}-\frac{2x-6}{8}\big{]},&5\leq
x<6;\\\ \frac{3}{4},&x=6;\\\
1-\frac{(8-x)^{2}}{16}+\frac{1}{20}\big{[}\frac{1}{4}-\frac{(8-x)^{2}}{16}\big{]},&6<x\leq
8-\sqrt{2};\\\
1-\frac{(8-x)^{2}}{16}+\frac{1}{20}\frac{(8-x)^{2}}{16},&8-\sqrt{2}\leq
x<8;\\\ 1,&x\geq 8.\\\ \end{array}\right.$

Similar to earlier, Figures 6,7, and 8 show the reduced single-fold UD of a
trapezoidal two-fold UV $\tilde{\xi}=\mathcal{TRA}(2,4,6,8;0.5,0.6)$ using
optimistic, pessimistic, and expected value criteria. Figures 6 and 7 in
particular are presented as solid figures because they are essentially bi-
variate functions with respect to $x$ and $\alpha$. For different values of
$\alpha\in(0,1)$, the reduced UDs via optimistic and pessimistic value
criteria are shown in Figures 9 and 10 in 2D pictures for clarity. It is
observed that for each fixed $\alpha$, the related curve satisfies the
conditions in Theorem 2.1, corresponding to a UD.

Figure 6: Reduced single-fold UD of a trapezoidal two-fold UD via $\alpha$
optimistic value criteria. Figure 7: Reduced single-fold UD of a trapezoidal
two-fold UD via $\alpha$ pessimistic value criteria. Figure 8: Reduced single-
fold UD of a trapezoidal two-fold UD via expected value criteria. Figure 9:
Reduced single-fold UDs of a trapezoidal two-fold UD via optimistic value
criteria for different values of $\alpha$. Figure 10: Reduced single-fold UDs
of a trapezoidal two-fold UD via pessimistic value criteria for different
values of $\alpha$.

##  4  Geometric Programming Problem with Two-fold Uncertain Coefficients

The conventional GP model assumes that the coefficients are precise and exact.
However, in the real-world GP model, the coefficients may be imprecise and
ambiguous. To overcome the difficulty, we propose to solve the GP problem in
an uncertain environment with the coefficients being triangular and
trapezoidal two-fold UVs. To do so, first, we develop the conventional GP
problem into a GP problem with coefficients that are two-fold UVs. In an
uncertain environment, the GP problem can be formulated as

$\displaystyle\min\quad\tilde{f}_{0}{(\textbf{x})}=\sum\limits_{i=1}^{N_{0}}{\tilde{\beta}_{i0}}^{2}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}$
(1) s.t.
$\displaystyle\qquad\tilde{f}_{k}{(\textbf{x})}=\sum\limits_{i=1}^{N_{k}}{\tilde{\beta}_{ik}}^{2}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}\leq
1,k=1,2,\ldots K,$

where the coefficients ${\tilde{\beta}_{ik}}^{2}$ are independent two-fold
UVs, $x_{j}>0$, $\alpha_{kij}\in\mathbb{R},\forall i,j,k$.

The goal of this study is to create the equivalent deterministic form of
Problem 1. To do so, we must first convert two-fold UVs
${\tilde{\beta}_{ik}}^{2}$ into single-fold UVs using optimistic, pessimistic,
and expected value criteria. If ${\tilde{\beta}_{ik}}^{2}$ is triangular two-
fold UV, then it can be reduced into single-fold UV via $\alpha$ optimistic,
$\alpha$ pessimistic, and expected value criteria based on Theorem 3.5,
Theorem 3.6, and Theorem 3.7, respectively. Similarly, if
${\tilde{\beta}_{ik}}^{2}$ is trapezoidal two-fold UV, then it can be reduced
into single-fold UV via $\alpha$ optimistic, $\alpha$ pessimistic, and
expected value criteria based on Theorem 3.8, Theorem 3.9, and Theorem 3.10,
respectively. Let ${\tilde{\beta}_{ik}}^{1}$ be the reduced single-fold UV
with the reduced single-fold UD $\Phi_{\tilde{\beta}_{ik}^{1}}$. After
reducing ${\tilde{\beta}_{ik}}^{2}$ into ${\tilde{\beta}_{ik}}^{1}$, we
consider Problem 1 in a chance-constrained uncertain-based framework. In this
framework, the problem is as follows.

$\displaystyle\min\quad\mathbb{E}\bigg{[}\sum\limits_{i=1}^{N_{0}}{\tilde{\beta}_{i0}}^{1}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}\bigg{]}$
(2) s.t.
$\displaystyle\qquad\mathcal{M}\bigg{(}\sum\limits_{i=1}^{N_{k}}{\tilde{\beta}_{ik}}^{1}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}\leq
1\bigg{)}\geq\gamma,k=1,2,\ldots K,$

where $\mathcal{M}$ is the single-fold uncertain measure, $\gamma\in(0,1)$ is
a predefined confidence level. To derive the equivalent deterministic form of
Problem 2, we use the following theorem developed by Liu Liu 2015 .

###### Theorem 4.1

Liu 2015  Assume the optimization problem in a chance-constrained uncertain
based framework as

$\displaystyle\min\quad\mathbb{E}\bigg{[}\tilde{f}_{0}{(\textbf{x};\xi_{1},\xi_{2},\ldots,\xi_{n})}\bigg{]}$
(3) s.t.
$\displaystyle\qquad\mathcal{M}\bigg{(}\tilde{f}_{k}{(\textbf{x};\xi_{1},\xi_{2},\ldots,\xi_{n})}\leq
0\bigg{)}\geq\gamma,k=1,2,\ldots K,$

where $\gamma\in(0,1)$ is a predefined confidence level. Let
$\xi_{1},\xi_{2},\ldots,\xi_{n}$ be independent UVs having regular UDs
$\Phi_{\xi_{1}},\Phi_{\xi_{2}},\ldots,\Phi_{\xi_{n}}$, respectively. If the
objective and constraint functions are strictly increasing with respect to
$\xi_{1},\xi_{2},\ldots,\xi_{n}$, then the deterministic form of Problem 3 is
as follows.

$\displaystyle\min\quad\int\limits_{0}^{1}\tilde{f}_{0}{\big{(}\textbf{x};\Phi_{\xi_{1}}^{-1}(\gamma),\Phi_{\xi_{2}}^{-1}(\gamma),\ldots,\Phi_{\xi_{n}}^{-1}(\gamma)\big{)}}d\gamma$
(4) s.t.
$\displaystyle\qquad\tilde{f}_{k}{\big{(}\textbf{x};\Phi_{\xi_{1}}^{-1}(\gamma),\Phi_{\xi_{2}}^{-1}(\gamma),\ldots,\Phi_{\xi_{n}}^{-1}(\gamma)\big{)}}\leq
0,k=1,2,\ldots K.$

Based on Theorem 4.1, Problem 2 becomes in a deterministic form as

$\displaystyle\min\quad\sum\limits_{i=1}^{N_{0}}\beta_{i0}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}$
(5) s.t.
$\displaystyle\qquad\sum\limits_{i=1}^{N_{k}}\beta_{ik}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}\leq
1,k=1,2,\ldots K,$

where $\gamma\in(0,1)$ is a predefined confidence level,
$\beta_{i0}=\int\limits_{0}^{1}\Phi_{\tilde{\beta}_{i0}^{1}}^{-1}(\gamma)d\gamma,i=1,2,\ldots,N_{0};$
$\beta_{ik}=\Phi_{\tilde{\beta}_{ik}^{1}}^{-1}(\gamma),i=1,2,\ldots,N_{k};k=1,2,\ldots,K.$
Problem 5 is a deterministic primal GP problem, where $N_{0}$ and $N_{k}$ are
the total number of terms of objective and $k^{th}$ constraint, respectively.
To define the dual problem, let $N=\sum\limits_{k=0}^{K}N_{k}$ be the total
numbers of terms presents in Problem 5, and $\delta_{ik}$ be the dual
variables such that
$\lambda_{k}=\sum\limits_{i=1}^{N_{k}}{\delta_{ik}},k=0,1,2,\ldots,K.$ Then
the dual problem is

$\displaystyle\max\quad
V(\delta)=\prod_{i=1}^{N}\bigg{(}\frac{\beta_{ik}}{{\delta_{ik}}}\bigg{)}^{{\delta_{ik}}}\bigg{(}\lambda_{k}\bigg{)}^{\lambda_{k}}$
(6) s.t.
$\displaystyle\qquad\sum\limits_{i=1}^{N_{0}}{\delta_{i0}}=\lambda_{0}=1,\quad(\text{Normality
condition})$
$\displaystyle\qquad\sum\limits_{i=1}^{N}{\delta_{ik}}{\alpha_{kij}}=0,j=1,2,\ldots,n.\quad(\text{Orthogonality
conditions})$

In a GP problem, the dual is solved, and then using the primal-dual
relationship, primal decision variables are found. Here is the relationship
between primal and dual problems.

Primal-dual Relationship: Due to the strong duality theorem Duffin 1967 ;
Duffin 1973 , here is how the primal and dual of a GP problem relate to each
other.

$\displaystyle\sum\limits_{j=1}^{n}{\alpha_{0ij}}\ln(x_{j})=\ln\bigg{(}\frac{{\delta_{i0}}f_{0}(\textbf{x})}{\beta_{i0}}\bigg{)},i=1,2,\ldots,N_{0},$
(7)
$\displaystyle\sum\limits_{j=1}^{n}{\alpha_{kij}}\ln(x_{j})=\ln\bigg{(}\frac{\delta_{ik}}{\lambda_{k}\beta_{ik}}\bigg{)},i=1,2,\ldots,N_{k},k=1,2,\ldots,K.$

###### Remark 8

Depending on $N$, we have the following two cases.

I.

If $N\geq n+1$, then Problem 6 is feasible as the number of equations is less
than or equal to the number of dual variables, which guarantees the existence
of a solution to the dual problem Beightler 1976 .

II.

If $N<n+1$, then Problem 6 is inconsistent as the number of equations is
greater than the number of dual variables. It guarantees that there is no
analytical solution to the dual problem. However, an approximate solution can
be found by the least square or linear programming method sinha 1987 .

##  5 Numerical Example

In this section, we provide a numerical example of a GP problem with two-fold
uncertainty. We consider the GP problem with two-fold uncertain coefficients
as

$\displaystyle\min\quad\tilde{f}_{0}(\textbf{x})=\tilde{\beta}_{10}^{2}x_{1}x_{2}+\tilde{\beta}_{20}^{2}x_{2}x_{3}+\tilde{\beta}_{30}^{2}x_{1}x_{3}$
(8) s.t.
$\displaystyle\qquad\tilde{f}_{1}(\textbf{x})=\frac{\tilde{\beta}_{11}^{2}}{x_{1}x_{2}x_{3}}\leq
1,$ $\displaystyle\qquad x_{1},x_{2},x_{3}>0.$

where
$\tilde{\beta}_{10}^{2},\tilde{\beta}_{20}^{2},\tilde{\beta}_{30}^{2},\tilde{\beta}_{11}^{2}$
are independent two-fold UVs. In this numerical example, we consider the
following two cases.

Case I: In this case, we assume that the coefficients of the objective and
constraints of Problem 8 are triangular two-fold UVs, which are given as
$\tilde{\beta}_{10}^{2}=\mathcal{TRI}(10,20,25;0.5,0.6)$,
$\tilde{\beta}_{20}^{2}=\mathcal{TRI}(30,40,50;0.4,0.6)$,
$\tilde{\beta}_{30}^{2}=\mathcal{TRI}(15,25,30;0.4,0.5)$,
$\tilde{\beta}_{11}^{2}=\mathcal{TRI}(6,8,9;0.5,0.7)$. For simplicity, we use
expected value reduction method to reduce two-fold UVs
$\tilde{\beta}_{10}^{2},\tilde{\beta}_{20}^{2},\tilde{\beta}_{30}^{2},\tilde{\beta}_{11}^{2}$
into single-fold UVs
$\tilde{\beta}_{10}^{1},\tilde{\beta}_{20}^{1},\tilde{\beta}_{30}^{1},\tilde{\beta}_{11}^{1}$,
respectively. Based on Theorem 3.7, UDs of reduced single-fold UVs
$\tilde{\beta}_{10}^{1},\tilde{\beta}_{20}^{1},\tilde{\beta}_{30}^{1},\tilde{\beta}_{11}^{1}$
are as follows.

$\Phi_{\tilde{\beta}_{10}^{1}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 10;\\\
\frac{(x-10)^{2}}{150}+\frac{1}{20}\frac{(x-10)^{2}}{150},&10<x\leq
10+5\sqrt{2};\\\
\frac{(x-10)^{2}}{150}+\frac{1}{20}\big{[}\frac{2}{3}-\frac{(x-10)^{2}}{150}\big{]},&10+5\sqrt{2}\leq
x<20;\\\ \frac{2}{3},&x=20;\\\
1-\frac{(25-x)^{2}}{75}+\frac{1}{20}\big{[}\frac{1}{3}-\frac{(25-x)^{2}}{75}\big{]},&20<x\leq
25-\frac{5}{\sqrt{2}};\\\
1-\frac{(25-x)^{2}}{75}+\frac{1}{20}\frac{(25-x)^{2}}{75},&25-\frac{5}{\sqrt{2}}\leq
x<25;\\\ 1,&x\geq 25;\\\ \end{array}\right.$

$\Phi_{\tilde{\beta}_{20}^{1}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 30;\\\
\frac{(x-30)^{2}}{200}+\frac{1}{10}\frac{(x-30)^{2}}{200},&30<x\leq
30+5\sqrt{2};\\\
\frac{(x-30)^{2}}{200}+\frac{1}{10}\big{[}\frac{1}{2}-\frac{(x-30)^{2}}{200}\big{]},&30+5\sqrt{2}\leq
x<40;\\\ \frac{1}{2},&x=40;\\\
1-\frac{(50-x)^{2}}{200}+\frac{1}{10}\big{[}\frac{1}{2}-\frac{(50-x)^{2}}{200}\big{]},&40<x\leq
50-5\sqrt{2};\\\
1-\frac{(50-x)^{2}}{200}+\frac{1}{10}\frac{(50-x)^{2}}{200},&50-5\sqrt{2}\leq
x<50;\\\ 1,&x\geq 50;\\\ \end{array}\right.$

$\Phi_{\tilde{\beta}_{30}^{1}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 15;\\\
\frac{(x-15)^{2}}{150}+\frac{1}{20}\frac{(x-15)^{2}}{150},&15<x\leq
15+5\sqrt{2};\\\
\frac{(x-15)^{2}}{150}+\frac{1}{20}\big{[}\frac{2}{3}-\frac{(x-15)^{2}}{150}\big{]},&15+5\sqrt{2}\leq
x<25;\\\ \frac{2}{3},&x=25;\\\
1-\frac{(30-x)^{2}}{75}+\frac{1}{20}\big{[}\frac{1}{3}-\frac{(30-x)^{2}}{75}\big{]},&25<x\leq
30-\frac{5}{\sqrt{2}};\\\
1-\frac{(30-x)^{2}}{75}+\frac{1}{20}\frac{(30-x)^{2}}{75},&30-\frac{5}{\sqrt{2}}\leq
x<30;\\\ 1,&x\geq 30;\\\ \end{array}\right.$

$\Phi_{\tilde{\beta}_{11}^{1}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 6;\\\
\frac{(x-6)^{2}}{6}+\frac{1}{10}\frac{(x-6)^{2}}{6},&6<x\leq 6+\sqrt{2};\\\
\frac{(x-6)^{2}}{6}+\frac{1}{10}\big{[}\frac{2}{3}-\frac{(x-6)^{2}}{6}\big{]},&6+\sqrt{2}\leq
x<8;\\\ \frac{2}{3},&x=8;\\\
1-\frac{(9-x)^{2}}{3}+\frac{1}{10}\big{[}\frac{1}{3}-\frac{(9-x)^{2}}{3}\big{]},&8<x\leq
9-\frac{1}{\sqrt{2}};\\\
1-\frac{(9-x)^{2}}{3}+\frac{1}{10}\frac{(9-x)^{2}}{3},&9-\frac{1}{\sqrt{2}}\leq
x<9;\\\ 1,&x\geq 9.\\\ \end{array}\right.$

Therefore, the inverse of those UDs are as follows.

$\Phi_{\tilde{\beta}_{10}^{1}}^{-1}(\gamma)=\left\\{\begin{array}[]{ll}10+\sqrt{\frac{1000}{7}\gamma},&0<\gamma\leq\frac{7}{20};\\\
10+\sqrt{\frac{3000}{19}\gamma-\frac{100}{19}},&\frac{7}{20}\leq\gamma\leq\frac{2}{3};\\\
25-\sqrt{\frac{1525}{21}-\frac{500}{7}\gamma},&\frac{2}{3}\leq\gamma\leq\frac{101}{120};\\\
25-\sqrt{\frac{1500}{19}-\frac{1500}{19}\gamma},&\frac{101}{120}\leq\gamma<1;\end{array}\right.$

$\Phi_{\tilde{\beta}_{20}^{1}}^{-1}(\gamma)=\left\\{\begin{array}[]{ll}30+\sqrt{\frac{2000}{11}\gamma},&0<\gamma\leq\frac{11}{40};\\\
30+\sqrt{\frac{2000}{9}\gamma-\frac{100}{9}},&\frac{11}{40}\leq\gamma\leq\frac{1}{2};\\\
50-\sqrt{\frac{2100}{11}-\frac{2000}{11}\gamma},&\frac{1}{2}\leq\gamma\leq\frac{31}{40};\\\
50-\sqrt{\frac{2000}{9}-\frac{2000}{9}\gamma},&\frac{31}{40}\leq\gamma<1;\end{array}\right.$

$\Phi_{\tilde{\beta}_{30}^{1}}^{-1}(\gamma)=\left\\{\begin{array}[]{ll}15+\sqrt{\frac{1000}{7}\gamma},&0<\gamma\leq\frac{7}{20};\\\
15+\sqrt{\frac{3000}{19}\gamma-\frac{100}{19}},&\frac{7}{20}\leq\gamma\leq\frac{2}{3};\\\
30-\sqrt{\frac{1525}{21}-\frac{500}{7}\gamma},&\frac{2}{3}\leq\gamma\leq\frac{101}{120};\\\
30-\sqrt{\frac{1500}{19}-\frac{1500}{19}\gamma},&\frac{101}{120}\leq\gamma<1;\end{array}\right.$

$\Phi_{\tilde{\beta}_{11}^{1}}^{-1}(\gamma)=\left\\{\begin{array}[]{ll}6+\sqrt{\frac{60}{11}\gamma},&0<\gamma\leq\frac{11}{30};\\\
6+\sqrt{\frac{20}{3}\gamma-\frac{4}{9}},&\frac{11}{30}\leq\gamma\leq\frac{2}{3};\\\
9-\sqrt{\frac{31}{11}-\frac{30}{11}\gamma},&\frac{2}{3}\leq\gamma\leq\frac{51}{60};\\\
9-\sqrt{\frac{10}{3}-\frac{10}{3}\gamma},&\frac{51}{60}\leq\gamma<1.\end{array}\right.$

Now, we have
$\beta_{10}=\int\limits_{0}^{1}\Phi_{\tilde{\beta}_{10}^{1}}^{-1}(\gamma)d\gamma,\beta_{20}=\int\limits_{0}^{1}\Phi_{\tilde{\beta}_{20}^{1}}^{-1}(\gamma)d\gamma,\text{
and
}\beta_{30}=\int\limits_{0}^{1}\Phi_{\tilde{\beta}_{30}^{1}}^{-1}(\gamma)d\gamma.$
After calculation, we have $\beta_{10}\approx 18.252,\beta_{20}\approx
39.804,\text{ and }\beta_{30}\approx 23.252.$ Therefore, the equivalent
deterministic form of Problem 8 is

$\displaystyle\min\quad 18.252x_{1}x_{2}+39.804x_{2}x_{3}+23.252x_{1}x_{3}$
(9) s.t.
$\displaystyle\qquad\frac{\Phi_{\tilde{\beta}_{11}^{1}}^{-1}(\gamma)}{x_{1}x_{2}x_{3}}\leq
1,$ $\displaystyle\qquad x_{1},x_{2},x_{3}>0.$

where $\Phi_{\tilde{\beta}_{11}^{1}}^{-1}(\gamma)$ is given above. Here is the
dual of Problem 9 as follows.

$\displaystyle\max\quad
V(\delta)=\bigg{(}\frac{18.252}{{\delta_{1}}}\bigg{)}^{{\delta_{1}}}\bigg{(}\frac{39.804}{{\delta_{2}}}\bigg{)}^{{\delta_{2}}}\bigg{(}\frac{23.252}{{\delta_{3}}}\bigg{)}^{{\delta_{3}}}\bigg{(}\frac{\Phi_{\tilde{\beta}_{11}^{1}}^{-1}(\gamma)}{{\delta_{4}}}\bigg{)}^{{\delta_{4}}}\big{(}\delta_{4}\big{)}^{\delta_{4}}$
(10) s.t.
$\displaystyle\qquad\delta_{1}+\delta_{2}+\delta_{3}=1,\quad(\text{Normality
condition})$ $\displaystyle\qquad\delta_{1}+\delta_{3}-\delta_{4}=0,$
$\displaystyle\qquad\delta_{1}+\delta_{2}-\delta_{4}=0,\quad(\text{Orthogonality
conditions})$ $\displaystyle\qquad\delta_{2}+\delta_{3}-\delta_{4}=0.$

Solving Problem 10, we get dual solution. Consequently, by the strong duality
theorem, we obtain the primal decision variables and expected objective value.
We find the solutions for confidence levels $\gamma\in(0,1)$, which are given
in Table 1.

Table 1:  Optimal solutions under triangular two-fold uncertainty $\gamma$ | $x_{1}^{*}$ | $x_{2}^{*}$ | $x_{3}^{*}$ | $\delta_{1}^{*}$ | $\delta_{2}^{*}$ | $\delta_{3}^{*}$ | $\delta_{4}^{*}$ | $\mathbb{E}[\tilde{f}_{0}(\textbf{x}^{*})]$
---|---|---|---|---|---|---|---|---
0.1 | 2.930 | 1.712 | 1.344 | 0.333 | 0.333 | 0.333 | 0.667 | 274.632
0.2 | 2.974 | 1.737 | 1.364 | 0.333 | 0.333 | 0.333 | 0.667 | 282.857
0.3 | 3.006 | 1.756 | 1.379 | 0.333 | 0.333 | 0.333 | 0.667 | 289.113
0.4 | 3.035 | 1.773 | 1.392 | 0.333 | 0.333 | 0.333 | 0.667 | 294.700
0.5 | 3.063 | 1.789 | 1.405 | 0.333 | 0.333 | 0.333 | 0.667 | 300.156
0.6 | 3.088 | 1.804 | 1.416 | 0.333 | 0.333 | 0.333 | 0.667 | 304.971
0.7 | 3.109 | 1.816 | 1.425 | 0.333 | 0.333 | 0.333 | 0.667 | 309.107
0.8 | 3.128 | 1.828 | 1.435 | 0.333 | 0.333 | 0.333 | 0.667 | 313.064
0.9 | 3.156 | 1.844 | 1.447 | 0.333 | 0.333 | 0.333 | 0.667 | 318.663

Figure 11 shows the expected objective value with respect to confidence level
under triangular two-fold uncertainty based on Table 1. It is observed that
the expected objective value is increasing with respect to the confidence
level.

Figure 11: Expected objective value with respect to confidence level under
triangular two-fold uncertainty.

Case II: In this case, we assume that the coefficients of the objective and
constraints of Problem 8 are trapezoidal two-fold UVs, which are given as
$\tilde{\beta}_{10}^{2}=\mathcal{TRA}(10,15,20,25;0.5,0.6)$,
$\tilde{\beta}_{20}^{2}=\mathcal{TRA}(30,40,50,60;0.4,0.6)$,
$\tilde{\beta}_{30}^{2}=\mathcal{TRA}(15,20,25,30;0.4,0.5)$,
$\tilde{\beta}_{11}^{2}=\mathcal{TRA}(6,7,8,9;0.5,0.7)$. Similar to earlier
case, we use expected value reduction method to reduce two-fold UVs
$\tilde{\beta}_{10}^{2},\tilde{\beta}_{20}^{2},\tilde{\beta}_{30}^{2},\tilde{\beta}_{11}^{2}$
into single-fold UVs
$\tilde{\beta}_{10}^{1},\tilde{\beta}_{20}^{1},\tilde{\beta}_{30}^{1},\tilde{\beta}_{11}^{1}$,
respectively. Based on Theorem 3.10, UDs of reduced single-fold UVs
$\tilde{\beta}_{10}^{1},\tilde{\beta}_{20}^{1},\tilde{\beta}_{30}^{1},\tilde{\beta}_{11}^{1}$
are as follows.

$\Phi_{\tilde{\beta}_{10}^{1}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 10;\\\
\frac{(x-10)^{2}}{100}+\frac{1}{20}\frac{(x-10)^{2}}{100},&10<x\leq
10+\frac{5}{\sqrt{2}};\\\
\frac{(x-10)^{2}}{100}+\frac{1}{20}\big{[}\frac{1}{4}-\frac{(x-10)^{2}}{100}\big{]},&10+\frac{5}{\sqrt{2}}\leq
x<15;\\\ \frac{1}{4},&x=15;\\\
\frac{2x-25}{20}+\frac{1}{20}\big{[}\frac{2x-25}{20}-\frac{1}{4}\big{]},&15<x\leq\frac{35}{2};\\\
\frac{2x-25}{20}+\frac{1}{20}\big{[}\frac{3}{4}-\frac{2x-25}{20}\big{]},&\frac{35}{2}\leq
x<20;\\\ \frac{3}{4},&x=20;\\\
1-\frac{(25-x)^{2}}{100}+\frac{1}{20}\big{[}\frac{1}{4}-\frac{(25-x)^{2}}{100}\big{]},&20<x\leq
25-\frac{5}{\sqrt{2}};\\\
1-\frac{(25-x)^{2}}{100}+\frac{1}{20}\frac{(25-x)^{2}}{100},&25-\frac{5}{\sqrt{2}}\leq
x<25;\\\ 1,&x\geq 25;\\\ \end{array}\right.$

$\Phi_{\tilde{\beta}_{20}^{1}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 30;\\\
\frac{(x-30)^{2}}{400}+\frac{1}{10}\frac{(x-30)^{2}}{400},&30<x\leq
30+5\sqrt{2};\\\
\frac{(x-30)^{2}}{400}+\frac{1}{10}\big{[}\frac{1}{4}-\frac{(x-30)^{2}}{400}\big{]},&30+5\sqrt{2}\leq
x<40;\\\ \frac{1}{4},&x=40;\\\
\frac{2x-70}{40}+\frac{1}{10}\big{[}\frac{2x-70}{40}-\frac{1}{4}\big{]},&40<x\leq
45;\\\
\frac{2x-70}{40}+\frac{1}{10}\big{[}\frac{3}{4}-\frac{2x-70}{40}\big{]},&45\leq
x<50;\\\ \frac{3}{4},&x=50;\\\
1-\frac{(60-x)^{2}}{400}+\frac{1}{10}\big{[}\frac{1}{4}-\frac{(60-x)^{2}}{400}\big{]},&50<x\leq
60-5\sqrt{2};\\\
1-\frac{(60-x)^{2}}{400}+\frac{1}{10}\frac{(60-x)^{2}}{400},&60-5\sqrt{2}\leq
x<60;\\\ 1,&x\geq 60;\\\ \end{array}\right.$

$\Phi_{\tilde{\beta}_{30}^{1}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 15;\\\
\frac{(x-15)^{2}}{100}+\frac{1}{20}\frac{(x-15)^{2}}{100},&15<x\leq
15+\frac{5}{\sqrt{2}};\\\
\frac{(x-15)^{2}}{100}+\frac{1}{20}\big{[}\frac{1}{4}-\frac{(x-15)^{2}}{100}\big{]},&15+\frac{5}{\sqrt{2}}\leq
x<20;\\\ \frac{1}{4},&x=20;\\\
\frac{2x-35}{20}+\frac{1}{20}\big{[}\frac{2x-35}{20}-\frac{1}{4}\big{]},&20<x\leq\frac{45}{2};\\\
\frac{2x-35}{20}+\frac{1}{20}\big{[}\frac{3}{4}-\frac{2x-35}{20}\big{]},&\frac{45}{2}\leq
x<25;\\\ \frac{3}{4},&x=25;\\\
1-\frac{(30-x)^{2}}{100}+\frac{1}{20}\big{[}\frac{1}{4}-\frac{(30-x)^{2}}{100}\big{]},&25<x\leq
30-\frac{5}{\sqrt{2}};\\\
1-\frac{(30-x)^{2}}{100}+\frac{1}{20}\frac{(30-x)^{2}}{100},&30-\frac{5}{\sqrt{2}}\leq
x<30;\\\ 1,&x\geq 30;\\\ \end{array}\right.$

$\Phi_{\tilde{\beta}_{11}^{1}}(x)=\left\\{\begin{array}[]{ll}0,&x\leq 6;\\\
\frac{(x-6)^{2}}{4}+\frac{1}{10}\frac{(x-6)^{2}}{4},&6<x\leq
6+\frac{1}{\sqrt{2}};\\\
\frac{(x-6)^{2}}{4}+\frac{1}{10}\big{[}\frac{1}{4}-\frac{(x-6)^{2}}{4}\big{]},&6+\frac{1}{\sqrt{2}}\leq
x<7;\\\ \frac{1}{4},&x=7;\\\
\frac{2x-13}{4}+\frac{1}{10}\big{[}\frac{2x-13}{4}-\frac{1}{4}\big{]},&7<x\leq\frac{15}{2};\\\
\frac{2x-13}{4}+\frac{1}{10}\big{[}\frac{3}{4}-\frac{2x-13}{4}\big{]},&\frac{15}{2}\leq
x<8;\\\ \frac{3}{4},&x=8;\\\
1-\frac{(9-x)^{2}}{4}+\frac{1}{10}\big{[}\frac{1}{4}-\frac{(9-x)^{2}}{4}\big{]},&8<x\leq
9-\frac{1}{\sqrt{2}};\\\
1-\frac{(9-x)^{2}}{4}+\frac{1}{10}\frac{(9-x)^{2}}{4},&9-\frac{1}{\sqrt{2}}\leq
x<9;\\\ 1,&x\geq 9.\\\ \end{array}\right.$

Therefore, the inverse of those UDs are as follows.

$\Phi_{\tilde{\beta}_{10}^{1}}^{-1}(\gamma)=\left\\{\begin{array}[]{ll}10+\sqrt{\frac{2000}{21}\gamma},&0<\gamma\leq\frac{21}{160};\\\
10+\sqrt{\frac{2000}{19}\gamma-\frac{25}{19}},&\frac{21}{160}\leq\gamma\leq\frac{1}{4};\\\
\frac{200}{21}\gamma+\frac{265}{21},&\frac{1}{4}\leq\gamma\leq\frac{41}{80};\\\
\frac{200}{19}\gamma+\frac{230}{19},&\frac{41}{80}\leq\gamma\leq\frac{3}{4};\\\
25-\sqrt{\frac{675}{7}-\frac{200}{21}\gamma},&\frac{3}{4}\leq\gamma\leq\frac{141}{160};\\\
25-\sqrt{\frac{2000}{19}-\frac{2000}{19}\gamma},&\frac{141}{160}\leq\gamma<1;\end{array}\right.$

$\Phi_{\tilde{\beta}_{20}^{1}}^{-1}(\gamma)=\left\\{\begin{array}[]{ll}30+\sqrt{\frac{4000}{11}\gamma},&0<\gamma\leq\frac{11}{80};\\\
30+\sqrt{\frac{4000}{9}\gamma-\frac{100}{9}},&\frac{11}{80}\leq\gamma\leq\frac{1}{4};\\\
\frac{200}{11}\gamma+\frac{390}{11},&\frac{1}{4}\leq\gamma\leq\frac{21}{40};\\\
\frac{200}{9}\gamma+\frac{100}{3},&\frac{21}{40}\leq\gamma\leq\frac{3}{4};\\\
60-\sqrt{\frac{4100}{11}-\frac{4000}{11}\gamma},&\frac{3}{4}\leq\gamma\leq\frac{71}{80};\\\
60-\sqrt{\frac{4000}{9}-\frac{4000}{9}\gamma},&\frac{71}{80}\leq\gamma<1;\end{array}\right.$

$\Phi_{\tilde{\beta}_{30}^{1}}^{-1}(\gamma)=\left\\{\begin{array}[]{ll}15+\sqrt{\frac{2000}{21}\gamma},&0<\gamma\leq\frac{21}{160};\\\
15+\sqrt{\frac{2000}{19}\gamma-\frac{25}{19}},&\frac{21}{160}\leq\gamma\leq\frac{1}{4};\\\
\frac{200}{21}\gamma+\frac{370}{21},&\frac{1}{4}\leq\gamma\leq\frac{41}{80};\\\
\frac{200}{19}\gamma+\frac{325}{19},&\frac{41}{80}\leq\gamma\leq\frac{3}{4};\\\
30-\sqrt{\frac{675}{7}-\frac{200}{21}\gamma},&\frac{3}{4}\leq\gamma\leq\frac{141}{160};\\\
30-\sqrt{\frac{2000}{19}-\frac{2000}{19}\gamma},&\frac{141}{160}\leq\gamma<1;\end{array}\right.$

$\Phi_{\tilde{\beta}_{11}^{1}}^{-1}(\gamma)=\left\\{\begin{array}[]{ll}6+\sqrt{\frac{40}{11}\gamma},&0<\gamma\leq\frac{11}{80};\\\
6+\sqrt{\frac{40}{9}\gamma-\frac{1}{9}},&\frac{11}{80}\leq\gamma\leq\frac{1}{4};\\\
\frac{20}{11}\gamma+\frac{72}{11},&\frac{1}{4}\leq\gamma\leq\frac{21}{40};\\\
\frac{20}{9}\gamma+\frac{19}{3},&\frac{21}{40}\leq\gamma\leq\frac{3}{4};\\\
9-\sqrt{\frac{41}{11}-\frac{40}{11}\gamma},&\frac{3}{4}\leq\gamma\leq\frac{71}{80};\\\
9-\sqrt{\frac{40}{9}-\frac{40}{9}\gamma},&\frac{71}{80}\leq\gamma<1.\end{array}\right.$

Similar to earlier case, we have
$\beta_{10}=\int\limits_{0}^{1}\Phi_{\tilde{\beta}_{10}^{1}}^{-1}(\gamma)d\gamma,\beta_{20}=\int\limits_{0}^{1}\Phi_{\tilde{\beta}_{20}^{1}}^{-1}(\gamma)d\gamma,\text{
and
}\beta_{30}=\int\limits_{0}^{1}\Phi_{\tilde{\beta}_{30}^{1}}^{-1}(\gamma)d\gamma.$
After calculation, we have $\beta_{10}\approx 17.745,\beta_{20}\approx
44.779,\text{ and }\beta_{30}\approx 21.775.$ Therefore, the equivalent
deterministic form of Problem 8 is

$\displaystyle\min\quad 17.745x_{1}x_{2}+44.779x_{2}x_{3}+21.775x_{1}x_{3}$
(11) s.t.
$\displaystyle\qquad\frac{\Phi_{\tilde{\beta}_{11}^{1}}^{-1}(\gamma)}{x_{1}x_{2}x_{3}}\leq
1,$ $\displaystyle\qquad x_{1},x_{2},x_{3}>0.$

where $\Phi_{\tilde{\beta}_{11}^{1}}^{-1}(\gamma)$ is given above. Here is the
dual of Problem 11 as follows.

$\displaystyle\max\quad
V(\delta)=\bigg{(}\frac{17.745}{{\delta_{1}}}\bigg{)}^{{\delta_{1}}}\bigg{(}\frac{44.779}{{\delta_{2}}}\bigg{)}^{{\delta_{2}}}\bigg{(}\frac{21.775}{{\delta_{3}}}\bigg{)}^{{\delta_{3}}}\bigg{(}\frac{\Phi_{\tilde{\beta}_{11}^{1}}^{-1}(\gamma)}{{\delta_{4}}}\bigg{)}^{{\delta_{4}}}\big{(}\delta_{4}\big{)}^{\delta_{4}}$
(12) $\displaystyle\text{s.t. the same normality and orthogonality conditions
given in Problem \ref{eq.10}}.$

Similarly, solving Problem 12, we get dual solution. Consequently, by the
strong duality theorem, we obtain the primal decision variables and expected
objective value. We find the solutions for confidence levels $\gamma\in(0,1)$,
which are given in Table 2.

Table 2:  Optimal solutions under trapezoidal two-fold uncertainty $\gamma$ | $x_{1}^{*}$ | $x_{2}^{*}$ | $x_{3}^{*}$ | $\delta_{1}^{*}$ | $\delta_{2}^{*}$ | $\delta_{3}^{*}$ | $\delta_{4}^{*}$ | $\mathbb{E}[\tilde{f}_{0}(\textbf{x}^{*})]$
---|---|---|---|---|---|---|---|---
0.1 | 3.248 | 1.579 | 1.287 | 0.333 | 0.333 | 0.333 | 0.667 | 273.098
0.2 | 3.293 | 1.601 | 1.305 | 0.333 | 0.333 | 0.333 | 0.667 | 280.738
0.3 | 3.326 | 1.617 | 1.318 | 0.333 | 0.333 | 0.333 | 0.667 | 286.393
0.4 | 3.354 | 1.631 | 1.329 | 0.333 | 0.333 | 0.333 | 0.667 | 291.273
0.5 | 3.382 | 1.645 | 1.340 | 0.333 | 0.333 | 0.333 | 0.667 | 296.112
0.6 | 3.414 | 1.660 | 1.353 | 0.333 | 0.333 | 0.333 | 0.667 | 301.699
0.7 | 3.447 | 1.676 | 1.366 | 0.333 | 0.333 | 0.333 | 0.667 | 307.496
0.8 | 3.476 | 1.690 | 1.378 | 0.333 | 0.333 | 0.333 | 0.667 | 312.825
0.9 | 3.510 | 1.707 | 1.391 | 0.333 | 0.333 | 0.333 | 0.667 | 318.927

In Figure 12, we show the expected objective value with respect to confidence
level under trapezoidal two-fold uncertainty based on Table 2. Here, too, the
expected objective value increases with confidence level.

Figure 12: Expected objective value with respect to confidence level under
trapezoidal two-fold uncertainty.

##  6 Conclusion

GP is a powerful technique for solving nonlinear optimization problems.
Conventional GP problems assume that the parameters are precise and exact.
However, in the real-world GP problem, the parameters may be imprecise and
ambiguous. In this paper, we study the GP problem with the coefficients as UVs
in an uncertain environment. We propose the definitions of triangular and
trapezoidal two-fold UVs. We develop three reduction methods: optimistic,
pessimistic, and expected value criteria, to reduce triangular and trapezoidal
two-fold UVs into single-fold UVs. We apply those reduction methods to a GP
problem with triangular and trapezoidal two-fold uncertainty. We show how the
GP problem with two-fold uncertainty is transformed into its equivalent
deterministic form. Finally, we provide the solution to a numerical example of
a GP problem under triangular and trapezoidal two-fold uncertainty to show the
efficiency and effectiveness of the procedure. In the numerical example, we
use only the expected value reduction method to reduce two-fold uncertainty
into single-fold uncertainty. One may also use optimistic and pessimistic
reduction methods.

For application purposes, one may consider solid transportation problems and
the inventory model under triangular and trapezoidal two-fold uncertainty for
further research.

Acknowledgement The first author is thankful to CSIR for financial support of
this work through file No: 09\1059(0027)\2019-EMR-I.
Data availability  Data sharing not applicable to this article as no datasets
were generated. A random data set is taken for the example and the inventory
model given in this paper.
Conflict of interests The authors have no relevant financial or non financial
interests to disclose. The authors declare that they have no conflict of
interests.

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