∎

11institutetext: Tapas Mondal* (Corresponding author) 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>
# Uncertain random geometric programming problems

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

(Received: date / Accepted: date)

###### Abstract

In this paper, we introduce a deterministic formulation for the geometric
programming problem, wherein the coefficients are represented as independent
linear-normal uncertain random variables. To address the challenges posed by
this combination of uncertainty and randomness, we introduce the concept of an
uncertain random variable and present a novel framework known as the linear-
normal uncertain random variable. Our main focus in this work is the
development of three distinct transformation techniques: the optimistic value
criteria, pessimistic value criteria, and expected value criteria. These
approaches allow us to convert a linear-normal uncertain random variable into
a more manageable random variable. This transition facilitates the
transformation from an uncertain random geometric programming problem to a
stochastic geometric programming problem. Furthermore, we provide insights
into an equivalent deterministic representation of the transformed geometric
programming problem, enhancing the clarity and practicality of the
optimization process. To demonstrate the effectiveness of our proposed
approach, we present a numerical example.

###### Keywords:

Stochastic programming Uncertainty modelling Linear-normal uncertain random
variable Geometric programming

###### MSC:

90C15 90C30 90C46 90C47 49K45

##  1 Introduction

Geometric programming (GP) is a highly efficient approach for addressing
nonlinear optimization problems when the objective and constraint functions
are in posynomial form. In 1967, Duffin et al. Duffin 1967  first introduced
the fundamental theories of the GP problem. In most cases, the GP method is
employed for objective and constraint functions, which are in posynomial form.
When all parameters in the GP problem are positive, except for the exponents,
it is referred to as the posynomial problem. The GP approach transforms the
original problem into a dual problem, making it more readily solvable.

In modern days, 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 classical GP problem is considered under the assumption that parameters
are well-defined and fixed. Many researchers Avriel 1975 ; Beightler 1976 ;
Duffin 1967 ; Duffin 1973 ; Fang 1988 ; Kortanek 1992 ; Kortanek 1997 ;
Maranas 1997 ; Rajgopal 1990 ; Rajgopal 1992 ; Rajgopal 2002  have devised
effective and functional algorithms designed for this traditional GP problem,
where parameters are exact and specific. In reality, though, the values
associated with GP parameters often lack certainty and precision.
Consequently, various approaches have been crafted to address the challenge of
GP in scenarios where parameter values are uncertain and less precise. Avriel
and Wilde Avriel 1969 , for instance, introduced a new dimension with the
stochastic GP problem, wherein they explored a GP scenario featuring objective
and constraint function coefficients as nonnegative random variables. Dupačová
Dupačová 2010  developed a metal cutting model in GP form and presented
stochastic sensivity analysis. In 2016, Liu et al. Liu 2016  solved the
stochastic GP problem with joint probabilistic constraints. Also, Liu et al.
Liu 2020  came up with a new way to solve the joint rectangular chance-
constrained GP problem with random parameters that are elliptically
distributed and independent from each other. In 2021, Shiraz et al. Shiraz
2021  solved the stochastic GP problem with joint chance constraints based on
copula theory. In this article, the coefficients of each term in the objective
and constraint functions of the GP problem are taken as dependent random
variables. Furthermore, the GP problem has been developed when the parameters
are taken as interval types. Liu Liu 2006  developed a technique for
determining the possible range of objective values in posynomial GP problems
when coefficients and exponents have varying intervals. Additionally,
Mahapatra and Mandal Mahapatra 2012  tackled the GP problem using interval-
valued coefficients within a parametric framework.

Over the past few decades, the GP problem has been developed in fuzzy
environments. In 1993, Cao Cao 1993  extended the GP problem to 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. 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 uncertainty
distributions. 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 distributions. In addition, the same authors Mondal 2023  proposed
triangular and trapezoidal two-fold uncertainty distributions and developed
three reduction methods to reduce two-fold uncertainty distributions into
single-fold uncertainty distributions. Chassien and Goerigk Chassein 2019
introduced a robust GP problem with polyhedral and interval uncertainty. In
2023, Fontem Fontem 2023  demonstrated how to set bounds for a worst-case
chance-constrained GP problem with random exponent parameters.

It is observed that the GP problem is taken into account under uncertainty
distributions whenever the parameters of uncertainty distributions are taken
in deterministic form. However, in reality, the parameters of an uncertainty
distribution are not always deterministic. So the obvious question is: how can
we deal with an uncertain variable when its parameters are random variables
instead of constant values? If the parameters of an uncertain variable are
random variables instead of constant values, then it is called an uncertain
random variable. In 2013, Liu Liu 2013a  proposed an uncertain random variable
that is a mixture of uncertainty and randomness. Further, the same author Liu
2013b  applied the concept of uncertain random variables to an optimization
problem called uncertain random programming. In 2021, Liu et al. Liu 2021
developed portfolio optimization for uncertain random returns. In addition,
Chen et al. Chen 2021  and Li et al. Li 2022  solved portfolio optimization
with uncertain random parameters. Zhou et al. Zhou 2014  introduced multi-
objective optimization in uncertain random environments. Wang et al. Wang 2015
developed an inventory model with uncertain random demand. Ke et al. Ke 2015
solved a decentralized decision-making production control problem with
uncertain random parameters. In 2018, Qin Qin 2018  introduced uncertain goal
programming. Gao et al. Gao 2018b  proposed the concept of an uncertain random
variable in a complex form. Ahmadzade et al. Ahmadzade 2020  and Gao et al.
Gao 2018a  introduced the convergence concept of distribution for an uncertain
random variable.

In most of the optimization problems with uncertain random parameters, the
authors considered some of the parameters to be uncertain variables and some
to be random variables. From the literature survey, it is observed that there
has been a lot of research on the crisp and fuzzy GP problem. In addition, an
optimization problem is developed when some of parameters are uncertain
variables and some of random variables. To the best of our knowledge, no
previous work on the GP problem with uncertain random coefficients has been
done. As a result, we attempt to investigate the GP problem with uncertain
random coefficients. The following are the main contributions we made to this
study:

I.

We introduce a novel deterministic formulation for the GP problem, where
coefficients are uniquely represented as independent linear-normal uncertain
random variables.

II.

To tackle uncertainty and randomness, we introduce the concept of an uncertain
random variable and propose a novel framework called the linear-normal
uncertain random variable.

III.

We develop three distinct transformation methods, each with its own unique
approach: the optimistic value criteria, pessimistic value criteria, and
expected value criteria. These methods are instrumental in converting complex
linear-normal uncertain random variables into more manageable, tractable
random variables.

IV.

The transformation methods enable us to convert a linear-normal uncertain
random variable into a more tractable random variable, facilitating the
transition from an uncertain random GP problem to a stochastic GP problem.

V.

We provide insights into an equivalent deterministic representation of the
transformed GP problem, adding clarity and practicality to the optimization
process.

VI.

To demonstrate the effectiveness and real-world applicability of our proposed
approach, we present a comprehensive numerical example, offering tangible
evidence of its efficacy in addressing GP problems.

In essence, our research presents a paradigm shift in addressing GP problems,
introducing innovative concepts and methods that enhance understanding,
clarity, and practicality in optimization processes.

The rest of the paper is organized as follows: A linear-normal uncertain
random variable is proposed in Section 2. Transformation methods are developed
in Section 3. The GP problem with linear-normal uncertain variable
coefficients is considered, and the deterministic formulation is developed in
Section 4. In Section 5, a numerical example is given. Finally, a conclusion
on this work is incorporated in Section 6.

##  2 Linear-normal uncertain random variable

In this section, we introduce the concept of a linear-normal uncertain random
variable. To begin, let us delve into some foundational ideas from both
probability theory and uncertainty theory.

###  2.1 Probability theory

Probability theory is a field of mathematics focused on exploring the
characteristics of unpredictable events. Within probability theory, the
concept of a random variable plays an important role. To grasp the essence of
a random variable, it is essential to familiarize ourselves with the notions
of probability measure and probability space.

###### Definition 1

Liu 2004  Consider the $\sigma-$algebra $L_{p}$ defined over a nonempty set
$\Omega$. To establish the concept of a random variable, we must initially
introduce the probability measure and probability space. The probability
measure, denoted as $Pr:L_{p}\rightarrow[0,1],$ adheres to the following
axioms:

I.

(Normality Axiom) $Pr\\{\Omega\\}=1$.

II.

(Nonnegativity Axiom) $Pr\\{A\\}\geq 0$ for any event $A\in L_{p}$.

III.

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

Furthermore, the trio denoted as $(\Omega,L_{p},Pr)$ constitutes the
probability space. The notion of a random variable proves highly valuable in
the context of probability description. Here, we provide the definition of a
random variable.

###### Definition 2

Liu 2004  A measurable function, denoted as
$\omega:(\Omega,L_{p},Pr)\to\mathbb{R}$, is referred to as a random variable.
In this context, $(\Omega,L_{p},Pr)$ represents the probability space, and
$\mathbb{R}$ is the set of real numbers. In simpler terms, for any Borel set
$B$, the set ${\omega\in B}$ is considered an event.

On the other hand, one can distinctly define a random variable through its
associated probability distribution, which holds essential information about
the random variable. In many instances, having knowledge of the probability
distribution is more than enough, obviating the need to know the random
variable directly. Below is the mathematical definition of probability
distribution.

###### Definition 3

Liu 2004  If $\Phi_{p}$ stands as the probability distribution associated to a
random variable denoted as $\omega$, then $\Phi_{p}$ is formally defined as

$\Phi_{p}(x)=Pr\\{\omega\leq x\\},\forall x\in\mathbb{R}.$

Additionally, the following theorem supports the requirement that is both
necessary and sufficient for a probability distribution function.

###### Theorem 2.1

Liu 2004  A function $\Phi_{p}:\mathbb{R}\to[0,1]$ qualifies as a probability
distribution if and only if it satisfies the conditions of being monotonically
increasing, right-continuous, and having the limits
$\underset{x\to-\infty}{\lim}\Phi_{p}(x)=0$ and
$\underset{x\to+\infty}{\lim}\Phi_{p}(x)=1$.

Normal random variables play an important role in order to define the linear-
normal uncertain random variable. Here is the definition of the normal random
variable.

###### Definition 4

Liu 2004  Let $\Phi_{p}$ be the probability distribution corresponding to a
random variable $\omega$. The random variable $\omega$ is said to be normal if
and only if $\Phi_{p}$ is defined as

$\Phi_{p}(x)=\frac{1}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{x-\mu}{\sigma\sqrt{2}}\bigg{)}\bigg{]},x\in\mathbb{R},$

where $\mu,\sigma$ are real numbers with $\sigma>0$, and
$\text{erf}(z)=\frac{2}{\sqrt{\pi}}\int\limits_{0}^{z}e^{-t^{2}}dt.$ If
$\omega$ is a normal random variable with the parameters $\mu$ and $\sigma$,
then we write it as $\omega\sim\mathcal{N}(\mu,\sigma).$

###### Remark 1

If $\mu=0$ and $\sigma=1$, then the random variable
$\omega\sim\mathcal{N}(0,1)$ is known as a standard normal random variable.
Its distribution function is

$\Phi(x)=\frac{1}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{x}{\sqrt{2}}\bigg{)}\bigg{]},x\in\mathbb{R}.$

###  2.2 Uncertainty theory

Uncertainty theory is an emerging mathematical discipline that employs
uncertain variables to depict the attributes of uncertainties and represent
degrees of belief. In order to introduce the concept of uncertain variables,
it is essential to grasp the notions of uncertain measure and uncertainty
space.

###### Definition 5

Liu 2015  Consider the $\sigma-$algebra $L_{u}$ defined over a nonempty set
$\Gamma$. To introduce the concept of an uncertain variable, we must initially
define the uncertain measure and the concept of an uncertainty space. The
uncertain measure is represented as the set function
$\mathcal{M}:L_{u}\rightarrow[0,1],$ which satisfies to the following axioms:

I.

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

II.

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

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 $\\{A_{i}\\}_{i=1}^{\infty}$.

Furthermore, the trio denoted as $(\Gamma,L_{u},\mathcal{M})$ constitutes the
uncertainty space. The concept of the uncertain variable is highly valuable in
the context of describing uncertainties. Below is the formal definition of the
uncertain variable.

###### Definition 6

Liu 2015  A measurable function denoted as
$\xi:(\Gamma,L_{u},\mathcal{M})\to\mathbb{R}$ is referred to as an uncertain
variable. In this context, $(\Gamma,L_{u},\mathcal{M})$ represents the
uncertainty space, and $\mathbb{R}$ corresponds to the set of real numbers.
Put simply, for any Borel set $B$, the set ${\xi\in B}$ is considered an
event.

Alternatively, an uncertain variable is identified through its associated
uncertainty distribution, which carries uncertain information regarding the
uncertain variable. In many instances, having knowledge of the uncertainty
distribution is more than sufficient, obviating the need to know the uncertain
variable directly. Here is the mathematical definition of uncertainty
distribution.

###### Definition 7

Liu 2015  If $\Phi_{u}$ represents the uncertainty distribution corresponding
to an uncertain variable $\xi$, then $\Phi_{u}$ is defined as

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

In addition, the requisite and complete condition for an uncertainty
distribution function is provided in the following theorem.

###### Theorem 2.2

Liu 2015  A function $\Phi_{u}:\mathbb{R}\to[0,1]$ qualifies as an uncertainty
distribution if and only if it displays monotonic increasing behavior,
excluding the cases where $\Phi_{u}(x)=0$ or $\Phi_{u}(x)=1.$

Describing uncertain information geometrically in a two-dimensional space
using the uncertainty distribution is straightforward. Real-world decision-
making systems often employ various uncertain variables, such as normal,
linear, zigzag, log-normal, triangular, and trapezoidal. Linear uncertain
variables are particularly significant in defining linear-normal uncertain
random variables. Here is the definition of a linear uncertain variable.

###### Definition 8

Liu 2015  Let $\Phi_{u}$ represents the uncertainty distribution corresponding
to an uncertain variable $\xi$. The uncertain variable $\xi$ is classified as
linear if and only if $\Phi_{u}$ is of the following form:

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

where $a$ and $b$ are real numbers with $a<b$. When $\xi$ is a linear
uncertain variable with parameters $a$ and $b$, we denote it as
$\xi\sim\mathcal{L}(a,b)$.

###  2.3 Uncertain random variable

An uncertain random variable is an uncertain variable whose parameters are
random variables instead of constant values. For better clarification, in a
linear uncertain variable $\xi\sim\mathcal{L}(a,b)$, the parameters $a$ and
$b$ are taken as constants. In this scenario, the obvious question is: how can
we deal with the linear uncertain variable $\xi\sim\mathcal{L}(a,b)$ if its
parameters $a$ and $b$ are random variables instead of constant values? In
order to describe this kind of variable, we need to define an uncertain random
variable.

###### Definition 9

An uncertain random variable $\tilde{\xi}$ is a function from the uncertainty
space $(\Gamma,L_{u},\mathcal{M})$ to the set of random variables such that
$Pr\\{\tilde{\xi}(\omega)\in B\\}$ is a measurable function of $\omega$ for
any Borel set $B$ of $\mathbb{R}.$

###### Remark 2

In broad terms, an uncertain random variable can be described as a measurable
function that operates within the uncertainty space and maps to the set of
random variables. In simpler terms, it is an element characterized by
uncertainty, which can take on values that belong to the category of random
variables.

In this research, we propose a linear-normal uncertain random variable. The
proposed definition is given below.

###### Definition 10

A linear-normal uncertain random variable is a linear uncertain variable whose
parameters are normal random variables. More precisely, the uncertain random
variable $\tilde{\xi}\sim\mathcal{L}(A,B)$ is said to be a linear-normal
uncertain random variable if and only if the parameters $A$ and $B$ are normal
random variables.

###### Remark 3

In Definition 10, the parameters $A$ and $B$ of the linear-normal uncertain
random variable $\tilde{\xi}\sim\mathcal{L}(A,B)$ are taken as independent
normal random variables.

##  3  Transformation of uncertain random variable to random variable

In this section, we demonstrate the process of converting an uncertain random
variable into a conventional random variable. To accomplish this, we employ
three critical value criteria: optimistic, pessimistic, and expected value
criteria. We begin by introducing the idea of critical values for uncertain
variables, which Liu Liu 2015  first introduced in 2015. Here is the
definition of these critical values.

###### Definition 11

Liu 2015 ; Yang 2015  If $\xi$ is an uncertain variable, 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 11 can be expressed using uncertainty distribution.
The subsequent theorem, commonly referred to as the measure inversion theorem,
proves to be highly valuable in this context.

###### Theorem 3.1

Liu 2015 ; Yang 2015  Consider an uncertain variable $\xi$ with an associated
uncertainty distribution $\Phi_{u}$. For any real number $r$, the following
relationships hold: $\mathcal{M}\\{\xi\leq r\\}=\Phi_{u}(r)\text{ and
}\mathcal{M}\\{\xi\geq r\\}=1-\Phi_{u}(r).$

Based on Theorem 3.1, Definition 11 can be redefined as

I.

$\xi_{\sup}(\alpha)=\sup\\{r|1-\Phi_{u}(r)\geq\alpha\\},\alpha\in(0,1);$

II.

$\xi_{\inf}(\alpha)=\inf\\{r|\Phi_{u}(r)\geq\alpha\\},\alpha\in(0,1);$

III.

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

###### Remark 4

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

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

###### Theorem 3.2

Liu 2015 ; Yang 2015  Let $\xi\sim\mathcal{L}(a,b)$ be a linear uncertain
variable. 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.

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

It is observed that if the parameters of an uncertain variable are random
variables instead of constant values, then the critical values of the
uncertain variable are random variables instead of constant values. Based on
this concept, we transform the uncertain random variable into a random
variable using the critical values of uncertainty. In particular, if
$\tilde{\xi}\sim\mathcal{L}(A,B)$ is a linear-normal uncertain random
variable, then based on Theorem 3.2, the transformed random variables using
optimistic, pessimistic, and expected value criteria are as follows:

I.

$\tilde{\xi}_{\sup}(\alpha)=\alpha A+(1-\alpha)B,\alpha\in(0,1);$

II.

$\tilde{\xi}_{\inf}(\alpha)=(1-\alpha)A+\alpha B,\alpha\in(0,1);$

III.

$\tilde{\xi}_{exp}=\frac{A+B}{2}.$

The main aim is to characterize the random variables
$\tilde{\xi}_{\sup}(\alpha),$ $\tilde{\xi}_{\inf}(\alpha),$ and
$\tilde{\xi}_{exp}.$ The characteristic function of a random variable is an
important concept and plays a powerful role in this regard. Here is the
definition of the characteristic function of a random variable.

###### Definition 12

Liu 2004  If $\omega$ is a random variable with probability distribution
$\Phi_{p}$, then its characteristic function is defined as
$\chi_{\omega}(t)=E[e^{it\omega}]=\int\limits_{-\infty}^{\infty}e^{itr}{\Phi^{\prime}_{p}}(r)dr,t\in\mathbb{R},i=\sqrt{-1}.$

Using the characteristic function, we characterize the transformed random
variables $\tilde{\xi}_{\sup}(\alpha),$ $\tilde{\xi}_{\inf}(\alpha),$ and
$\tilde{\xi}_{exp}.$ The following theorem is helpful in this regard.

###### Theorem 3.3

If $\tilde{\xi}\sim\mathcal{L}(A,B)$ is a linear-normal uncertain random
variable where $A\sim\mathcal{N}(\mu_{A},\sigma_{A})$ and
$B\sim\mathcal{N}(\mu_{B},\sigma_{B})$ are two independent normal random
variables, then the following results hold:

I.

$\tilde{\xi}_{\sup}(\alpha)\sim\mathcal{N}\big{(}\alpha\mu_{A}+(1-\alpha)\mu_{B},\sqrt{\alpha^{2}\sigma_{A}^{2}+(1-\alpha)^{2}\sigma_{B}^{2}}\big{)},\alpha\in(0,1);$

II.

$\tilde{\xi}_{\inf}(\alpha)\sim\mathcal{N}\big{(}(1-\alpha)\mu_{A}+\alpha\mu_{B},\sqrt{(1-\alpha)^{2}\sigma_{A}^{2}+\alpha^{2}\sigma_{B}^{2}}\big{)},\alpha\in(0,1);$

III.

$\tilde{\xi}_{exp}\sim\mathcal{N}\big{(}\frac{\mu_{A}+\mu_{B}}{2},\frac{\sqrt{\sigma_{A}^{2}+\sigma_{B}^{2}}}{2}\big{)}.$

###### Proof

We prove only (I). The proofs of (II) and (III) are similar to the proof of
(I).

We have $\tilde{\xi}_{\sup}(\alpha)=\alpha A+(1-\alpha)B,$ where
$A\sim\mathcal{N}(\mu_{A},\sigma_{A})$ and
$B\sim\mathcal{N}(\mu_{B},\sigma_{B})$ are two independent normal random
variables. So, the characteristic functions of $A$ and $B$ are
$\chi_{A}(t)=E[e^{itA}]$ and $\chi_{B}(t)=E[e^{itB}].$ Using Definition 12, we
get $\chi_{A}(t)=e^{i\mu_{A}t-\frac{1}{2}\sigma_{A}^{2}t^{2}}$ and
$\chi_{B}(t)=e^{i\mu_{B}t-\frac{1}{2}\sigma_{B}^{2}t^{2}}.$ Now, the
characteristic function of the random variable $\tilde{\xi}_{\sup}(\alpha)$ is
calculated as

$\displaystyle\chi_{\tilde{\xi}_{\sup}}(t;\alpha)$
$\displaystyle=E\big{[}e^{it\tilde{\xi}_{\sup}(\alpha)}\big{]}$
$\displaystyle=E\bigg{[}e^{it\big{(}\alpha A+(1-\alpha)B\big{)}}\bigg{]}$
$\displaystyle=E\big{[}e^{it\alpha A}\cdot e^{it(1-\alpha)B}\big{]}$
$\displaystyle=E\big{[}e^{it\alpha A}\big{]}E\big{[}e^{it(1-\alpha)B}\big{]}$
$\displaystyle=\chi_{A}(\alpha t)\cdot\chi_{B}((1-\alpha)t)$
$\displaystyle=e^{i\mu_{A}(\alpha t)-\frac{1}{2}\sigma_{A}^{2}(\alpha
t)^{2}}\cdot
e^{i\mu_{B}((1-\alpha)t)-\frac{1}{2}\sigma_{B}^{2}((1-\alpha)t)^{2}}$
$\displaystyle=e^{i(\alpha\mu_{A}+(1-\alpha)\mu_{B})t-\frac{1}{2}(\alpha^{2}\sigma_{A}^{2}+(1-\alpha)^{2}\sigma_{B}^{2})t^{2}}.$

Therefore, the characteristic function of $\tilde{\xi}_{\sup}(\alpha)$ is
$\chi_{\tilde{\xi}_{\sup}}(t;\alpha)=e^{i(\alpha\mu_{A}+(1-\alpha)\mu_{B})t-\frac{1}{2}(\alpha^{2}\sigma_{A}^{2}+(1-\alpha)^{2}\sigma_{B}^{2})t^{2}}.$
This shows that
$\tilde{\xi}_{\sup}(\alpha)\sim\mathcal{N}\big{(}\alpha\mu_{A}+(1-\alpha)\mu_{B},\sqrt{\alpha^{2}\sigma_{A}^{2}+(1-\alpha)^{2}\sigma_{B}^{2}}\big{)},$
which completes the proof.

It is observed that the parameters of the transformed random variables
$\tilde{\xi}_{\sup}(\alpha),$ $\tilde{\xi}_{\inf}(\alpha),$ and
$\tilde{\xi}_{exp}$ are known. So, we can visualize their corresponding
probability distribution functions. The probability distributions of
$\tilde{\xi}_{\sup}(\alpha),$ $\tilde{\xi}_{\inf}(\alpha),$ and
$\tilde{\xi}_{exp}$ are as follows:

I.

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\frac{1}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{x-\alpha\mu_{A}-(1-\alpha)\mu_{B}}{\sqrt{2\alpha^{2}\sigma_{A}^{2}+2(1-\alpha)^{2}\sigma_{B}^{2}}}\bigg{)}\bigg{]},x\in\mathbb{R},\alpha\in(0,1);$

II.

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\frac{1}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{x-(1-\alpha)\mu_{A}-\alpha\mu_{B}}{\sqrt{2(1-\alpha)^{2}\sigma_{A}^{2}+2\alpha^{2}\sigma_{B}^{2}}}\bigg{)}\bigg{]},x\in\mathbb{R},\alpha\in(0,1);$

III.

$\Phi_{\tilde{\xi}_{exp}}(x)=\frac{1}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{2x-\mu_{A}-\mu_{B}}{\sqrt{2\sigma_{A}^{2}+2\sigma_{B}^{2}}}\bigg{)}\bigg{]},x\in\mathbb{R}.$

To understand the efficiency of the transformation methods, we provide an
example of a linear-normal uncertain random variable below.

###### Example 1

If $\tilde{\xi}\sim\mathcal{LN}(A,B)$ is a linear-normal uncertain random
variable where $A\sim\mathcal{N}(3,1)$ and $B\sim\mathcal{N}(2,1)$ are two
independent normal random variables, then

I.

the transformed random variable via $\alpha$ optimistic value criteria is
$\tilde{\xi}_{\sup}(\alpha)\sim\mathcal{N}(\alpha+2,\sqrt{\alpha^{2}+(1-\alpha)^{2}})$
and its corresponding probability distribution is

$\Phi_{\tilde{\xi}_{\sup}}(x;\alpha)=\frac{1}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{x-\alpha-2}{\sqrt{2\alpha^{2}+2(1-\alpha)^{2}}}\bigg{)}\bigg{]},x\in\mathbb{R},\alpha\in(0,1);$

II.

the transformed random variable via $\alpha$ pessimistic value criteria is
$\tilde{\xi}_{\inf}(\alpha)\sim\mathcal{N}(3-\alpha,\sqrt{\alpha^{2}+(1-\alpha)^{2}})$
and its corresponding probability distribution is

$\Phi_{\tilde{\xi}_{\inf}}(x;\alpha)=\frac{1}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{x+\alpha-3}{\sqrt{2\alpha^{2}+2(1-\alpha)^{2}}}\bigg{)}\bigg{]},x\in\mathbb{R},\alpha\in(0,1);$

III.

the transformed random variable via expected value criteria is
$\tilde{\xi}_{exp}\sim\mathcal{N}(\frac{5}{2},\frac{\sqrt{2}}{2})$ and its
corresponding probability distribution is

$\Phi_{\tilde{\xi}_{exp}}(x)=\frac{1}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{2x-5}{2}\bigg{)}\bigg{]},x\in\mathbb{R}.$

Utilizing the optimistic, pessimistic, and expected value criteria, we depict
the transformed probability distribution of the linear-normal uncertain random
variable $\tilde{\xi}\sim\mathcal{LN}(A,B)$ in Figure 1, Figure 2, and Figure
3, correspondingly, where $A\sim\mathcal{N}(3,1)$ and $B\sim\mathcal{N}(2,1)$.
Figure 1 and Figure 2 are presented as solid illustrations because they
essentially represent bivariate functions involving both $x$ and $\alpha$. To
offer a clear visualization for different values of $\alpha\in(0,1)$, we
present the transformed probability distributions using the optimistic and
pessimistic value criteria in Figure 4 and Figure 5 as two-dimensional plots.

###### Remark 5

The optimistic value criteria transformation method for a fixed
$\alpha\in(0,1)$ is equivalent to the pessimistic value criteria
transformation method for $(1-\alpha)$. In particular, at $\alpha=0.5,$ the
optimistic value criteria and pessimistic value criteria transformation
methods are equivalent to the expected value criteria transformation method.

Figure 1: Transformed probability distribution of a linear-normal uncertain
random variable via $\alpha$ optimistic value criteria. Figure 2: Transformed
probability distribution of a linear-normal uncertain random variable via
$\alpha$ pessimistic value criteria. Figure 3: Transformed probability
distribution of a linear-normal uncertain random variable via expected value
criteria. Figure 4: Transformed probability distributions of a linear-normal
uncertain random variable via optimistic value criteria for different values
of $\alpha$. Figure 5: Transformed probability distributions of a linear-
normal uncertain random variable via pessimistic value criteria for different
values of $\alpha$.

##  4  Geometric programming problem with linear-normal uncertain random
variable coefficients

A conventional GP problem involves addressing a minimization problem where
both the objective function and the constraints exhibit posynomial
characteristics. Problems within the GP framework, in which all coefficients
and decision variables are positive except for the exponents, are categorized
as posynomial problems. A deterministic GP problem is outlined as

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

where $\beta_{ik}>0,x_{j}>0,\alpha_{kij}\in\mathbb{R},\forall i,j,k$. In this
context, we have $N_{0}$ representing the total number of terms in the
objective function, and $N_{k}$ representing the total number of terms in the
$k^{th}$ constraint of Problem 1. To formulate the dual for Problem 1, let us
introduce the variable $N$ as the total number of terms present in Problem 1,
given by $N=\sum\limits_{k=0}^{K}N_{k}$. Additionally, we use $\delta_{ik}$ as
the dual variables, and $\lambda_{k}$ as the sum of dual variables present in
the $k^{th}$ constraint, defined as
$\lambda_{k}=\sum\limits_{i=1}^{N_{k}}{\delta_{ik}}$, where
$k=0,1,2,\ldots,K$. Consequently, the dual problem for Problem 1 can be
expressed as

$\displaystyle\max\quad
V(\delta)=\prod_{i=1}^{N}\bigg{(}\frac{\beta_{ik}}{{\delta_{ik}}}\bigg{)}^{{\delta_{ik}}}\bigg{(}\lambda_{k}\bigg{)}^{\lambda_{k}}$
(2) 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 primal problem is converted into its dual problem, which
is easier to solve than the primal problem. The relationship between primal
and dual problems is useful to find the primal decision variables. The
relationship between primal and dual GP problems due to strong duality theorem
Duffin 1967 ; Duffin 1973  is as follows:

$\displaystyle\sum\limits_{j=1}^{n}{\alpha_{0ij}}\ln(x_{j})=\ln\bigg{(}\frac{{\delta_{i0}}f_{0}(\textbf{x})}{\beta_{i0}}\bigg{)},i=1,2,\ldots,N_{0},$
(3)
$\displaystyle\sum\limits_{j=1}^{n}{\alpha_{kij}}\ln(x_{j})=\ln\bigg{(}\frac{\delta_{ik}}{\lambda_{k}\beta_{ik}}\bigg{)},i=1,2,\ldots,N_{k},k=1,2,\ldots,K.$

###### Remark 6

In a GP problem, the total number of terms minus the total number of decision
variables minus one is known as the degree of difficulty. In Problem 1 and its
dual given in Problem 2, the degree of difficulty is $D=N-n-1$. Depending on
$D$, we have the following two cases.

I.

If $D\geq 0$, then Problem 2 is feasible. Because in this case, the number of
equations is less than or equal to the number of dual variables. This
guarantees the existence of a solution to Problem 2.

II.

On the other hand, if $D<0$, then Problem 2 is inconsistent. Because in this
case, 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,
Sinha and Biswal sinha 1987  found an approximate solution to the dual problem
with a negative degree of difficulty using the least squares method and the
linear programming method.

In this section, our main aim is to develop an equivalent deterministic form
of a GP problem in which the coefficients of the objective and constraint
functions are linear-normal uncertain random variables. First, we consider the
uncertain random GP problem as

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

where the coefficients ${\tilde{\beta}_{ik}}\sim\mathcal{LN}(A_{ik},B_{ik})$
are independent linear-normal uncertain random variables,
$A_{ik}\sim\mathcal{N}(\mu_{A_{ik}},\sigma_{A_{ik}})$ and
$B_{ik}\sim\mathcal{N}(\mu_{B_{ik}},\sigma_{B_{ik}})$ are independent normal
random variables, $x_{j}>0$, $\alpha_{kij}\in\mathbb{R},\forall i,j,k$. To
develop the equivalent deterministic form of Problem 4, we first transform
uncertain random variable coefficients
${\tilde{\beta}_{ik}}\sim\mathcal{LN}(A_{ik},B_{ik})$ into random variable
coefficients based on Theorem 3.3. For instance, Problem 4 is converted into
stochastic GP problems via optimistic, pessimistic, and expected value
criteria. Here are the transformations of Problem 4 into stochastic GP
problems.

I.

The transformation of Problem 4 into the stochastic GP problem via $\alpha$
optimistic value criteria is

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

where
${\tilde{\beta}}_{{ik}_{\sup}}(\alpha)\sim\mathcal{N}\big{(}\alpha\mu_{A_{ik}}+(1-\alpha)\mu_{B_{ik}},\sqrt{\alpha^{2}\sigma_{A_{ik}}^{2}+(1-\alpha)^{2}\sigma_{B_{ik}}^{2}}\big{)}$
are independent normal random variables, $\alpha\in(0,1)$.

II.

The transformation of Problem 4 into the stochastic GP problem via $\alpha$
pessimistic value criteria is

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

where
${\tilde{\beta}}_{{ik}_{\inf}}(\alpha)\sim\mathcal{N}\big{(}(1-\alpha)\mu_{A_{ik}}+\alpha\mu_{B_{ik}},\sqrt{(1-\alpha)^{2}\sigma_{A_{ik}}^{2}+\alpha^{2}\sigma_{B_{ik}}^{2}}\big{)}$
are independent normal random variables, $\alpha\in(0,1)$.

III.

The transformation of Problem 4 into the stochastic GP problem via expected
value criteria is

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

where
${\tilde{\beta}}_{{ik}_{exp}}\sim\mathcal{N}\bigg{(}\frac{\mu_{A_{ik}}+\mu_{B_{ik}}}{2},\frac{\sqrt{\sigma_{A_{ik}}^{2}+\sigma_{B_{ik}}^{2}}}{2}\bigg{)}$
are independent normal random variables.

We use probabilistic constraints with a tolerance level $\epsilon\in(0,0.5].$
Then Problem 5 becomes

$\displaystyle\min x_{0}$ (8) s.t. $\displaystyle\qquad
Pr\bigg{(}\sum\limits_{i=1}^{N_{k}}{\tilde{\beta}}_{{ik}_{\sup}}(\alpha)\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}\leq
1\bigg{)}\geq 1-\epsilon,k=1,2,\ldots K,$ $\displaystyle\qquad
Pr\bigg{(}\sum\limits_{i=1}^{N_{0}}{\tilde{\beta}}_{{i0}_{\sup}}(\alpha)\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}x_{0}^{-1}\leq
1\bigg{)}\geq 1-\epsilon,$

Problem 6 becomes

$\displaystyle\min\quad x_{0}$ (9) s.t. $\displaystyle\qquad
Pr\bigg{(}\sum\limits_{i=1}^{N_{k}}{\tilde{\beta}}_{{ik}_{\inf}}(\alpha)\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}\leq
1\bigg{)}\geq 1-\epsilon,k=1,2,\ldots K,$ $\displaystyle\qquad
Pr\bigg{(}\sum\limits_{i=1}^{N_{0}}{\tilde{\beta}}_{{i0}_{\inf}}(\alpha)\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}x_{0}^{-1}\leq
1\bigg{)}\geq 1-\epsilon,$

Problem 7 becomes

$\displaystyle\min\quad x_{0}$ (10) s.t. $\displaystyle\qquad
Pr\bigg{(}\sum\limits_{i=1}^{N_{k}}{\tilde{\beta}}_{{ik}_{exp}}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}\leq
1\bigg{)}\geq 1-\epsilon,k=1,2,\ldots K,$ $\displaystyle\qquad
Pr\bigg{(}\sum\limits_{i=1}^{N_{0}}{\tilde{\beta}}_{{i0}_{exp}}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}x_{0}^{-1}\leq
1\bigg{)}\geq 1-\epsilon.$

Now, the main focus is to find an equivalent deterministic form of a
probabilistic constraint. The following theorem is very useful in this regard.

###### Theorem 4.1

If $\xi_{l}\sim\mathcal{N}(\mu_{l},\sigma_{l})$ with $l=1,2,\ldots,L$ are
independent normal random variables, and $U_{l}$ with $l=1,2,\ldots,L$ are
nonnegative variables, then for every tolerance level $\epsilon\in(0,0.5]$,

$\displaystyle Pr\bigg{(}\sum\limits_{l=1}^{L}\xi_{l}U_{l}\leq 1\bigg{)}\geq
1-\epsilon$

is equivalent to

$\displaystyle\sum\limits_{l=1}^{L}\mu_{l}U_{l}+\Phi^{-1}(1-\epsilon)\sqrt{\sum\limits_{l=1}^{L}\sigma_{l}^{2}U_{l}^{2}}\leq
1,$

where $\Phi^{-1}(1-\epsilon)=\sqrt{2}\text{erf}^{-1}(1-2\epsilon)$ is the
inverse of the standard normal distribution function.

###### Proof

Let $Z=\bigg{(}\sum\limits_{l=1}^{L}\xi_{l}U_{l}\bigg{)}-1.$ Then we have
$\mu_{Z}=E[Z]=E\bigg{[}\bigg{(}\sum\limits_{l=1}^{L}\xi_{l}U_{l}\bigg{)}-1\bigg{]}=\bigg{(}\sum\limits_{l=1}^{L}E[\xi_{l}]U_{l}\bigg{)}-1=\sum\limits_{l=1}^{L}\mu_{l}U_{l}-1$
and
$\sigma_{Z}^{2}=var(Z)=var\bigg{[}\bigg{(}\sum\limits_{l=1}^{L}\xi_{l}U_{l}\bigg{)}-1\bigg{]}=\bigg{(}\sum\limits_{l=1}^{L}var[\xi_{l}]U_{l}^{2}\bigg{)}-0=\sum\limits_{l=1}^{L}\sigma_{l}^{2}U_{l}^{2}.$
Therefore, we have

$\displaystyle Pr\bigg{(}\sum\limits_{l=1}^{L}\xi_{l}U_{l}\leq 1\bigg{)}\geq
1-\epsilon$ $\displaystyle\Leftrightarrow Pr(Z\leq 0)\geq 1-\epsilon$
$\displaystyle\Leftrightarrow
Pr\bigg{(}\frac{Z-\mu_{Z}}{\sigma_{Z}}\leq-\frac{\mu_{Z}}{\sigma_{Z}}\bigg{)}\geq
1-\epsilon$
$\displaystyle\Leftrightarrow\Phi\bigg{(}-\frac{\mu_{Z}}{\sigma_{Z}}\bigg{)}\geq
1-\epsilon\qquad\bigg{[}\because\frac{Z-\mu_{Z}}{\sigma_{Z}}\sim\mathcal{N}(0,1)\bigg{]}$
$\displaystyle\Leftrightarrow-\frac{\mu_{Z}}{\sigma_{Z}}\geq\Phi^{-1}(1-\epsilon)$
$\displaystyle\Leftrightarrow\mu_{Z}+\sigma_{Z}\Phi^{-1}(1-\epsilon)\leq 0$
$\displaystyle\Leftrightarrow\sum\limits_{l=1}^{L}\mu_{l}U_{l}-1+\Phi^{-1}(1-\epsilon)\sqrt{\sum\limits_{l=1}^{L}\sigma_{l}^{2}U_{l}^{2}}\leq
0$
$\displaystyle\Leftrightarrow\sum\limits_{l=1}^{L}\mu_{l}U_{l}+\Phi^{-1}(1-\epsilon)\sqrt{\sum\limits_{l=1}^{L}\sigma_{l}^{2}U_{l}^{2}}\leq
1.$

This completes the proof.

Based on Theorem 4.1, we convert each probabilistic constraint into a
deterministic constraint. Therefore, converting probabilistic constraints into
deterministic constraints and, after simplification, the deterministic form of
Problem 8, Problem 9, and Problem 10 are derived in Problem 11, Problem 12,
and Problem 13, respectively. Problem 11 is

$\displaystyle\min\sum\limits_{i=1}^{N_{0}}\big{(}\alpha\mu_{A_{i0}}+(1-\alpha)\mu_{B_{i0}}\big{)}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}+\Phi^{-1}(1-\epsilon)\sqrt{\sum\limits_{i=1}^{N_{0}}\big{(}\alpha^{2}\sigma_{A_{i0}}^{2}+(1-\alpha)^{2}\sigma_{B_{i0}}^{2}\big{)}\prod\limits_{j=1}^{n}x_{j}^{2\alpha_{0ij}}}$
(11) s.t.
$\displaystyle\quad\sum\limits_{i=1}^{N_{k}}\big{(}\alpha\mu_{A_{ik}}+(1-\alpha)\mu_{B_{ik}}\big{)}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}+\Phi^{-1}(1-\epsilon)\sqrt{\sum\limits_{i=1}^{N_{k}}\big{(}\alpha^{2}\sigma_{A_{ik}}^{2}+(1-\alpha)^{2}\sigma_{B_{ik}}^{2}\big{)}\prod\limits_{j=1}^{n}x_{j}^{2\alpha_{kij}}}\leq
1,$ $\displaystyle\quad\text{where
}\alpha\in(0,1),\epsilon\in(0,0.5],k=1,2,\ldots K.$

Problem 12 is

$\displaystyle\min\sum\limits_{i=1}^{N_{0}}\big{(}(1-\alpha)\mu_{A_{i0}}+\alpha\mu_{B_{i0}}\big{)}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}+\Phi^{-1}(1-\epsilon)\sqrt{\sum\limits_{i=1}^{N_{0}}\big{(}(1-\alpha)^{2}\sigma_{A_{i0}}^{2}+\alpha^{2}\sigma_{B_{i0}}^{2}\big{)}\prod\limits_{j=1}^{n}x_{j}^{2\alpha_{0ij}}}$
(12) s.t.
$\displaystyle\quad\sum\limits_{i=1}^{N_{k}}\big{(}(1-\alpha)\mu_{A_{ik}}+\alpha\mu_{B_{ik}}\big{)}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}+\Phi^{-1}(1-\epsilon)\sqrt{\sum\limits_{i=1}^{N_{k}}\big{(}(1-\alpha)^{2}\sigma_{A_{ik}}^{2}+\alpha^{2}\sigma_{B_{ik}}^{2}\big{)}\prod\limits_{j=1}^{n}x_{j}^{2\alpha_{kij}}}\leq
1,$ $\displaystyle\quad\text{where
}\alpha\in(0,1),\epsilon\in(0,0.5],k=1,2,\ldots K.$

Problem 13 is

$\displaystyle\min\sum\limits_{i=1}^{N_{0}}\bigg{(}\frac{\mu_{A_{i0}}+\mu_{B_{i0}}}{2}\bigg{)}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{0ij}}+\frac{\Phi^{-1}(1-\epsilon)}{2}\sqrt{\sum\limits_{i=1}^{N_{0}}\big{(}\sigma_{A_{i0}}^{2}+\sigma_{B_{i0}}^{2}\big{)}\prod\limits_{j=1}^{n}x_{j}^{2\alpha_{0ij}}}$
(13) s.t.
$\displaystyle\quad\sum\limits_{i=1}^{N_{k}}\bigg{(}\frac{\mu_{A_{ik}}+\mu_{B_{ik}}}{2}\bigg{)}\prod\limits_{j=1}^{n}x_{j}^{\alpha_{kij}}+\frac{\Phi^{-1}(1-\epsilon)}{2}\sqrt{\sum\limits_{i=1}^{N_{k}}\big{(}\sigma_{A_{ik}}^{2}+\sigma_{B_{ik}}^{2}\big{)}\prod\limits_{j=1}^{n}x_{j}^{2\alpha_{kij}}}\leq
1,$ $\displaystyle\quad\text{where }\epsilon\in(0,0.5],k=1,2,\ldots K.$

##  5 Numerical example

In this section, we solve a numerical example to show the efficiency and
efficacy of the procedures. We consider a GP problem with linear-normal
uncertain random variable coefficients as

$\displaystyle\min\quad\tilde{f}_{0}(\textbf{x})=\tilde{\beta}_{10}x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}+\tilde{\beta}_{20}x_{1}x_{3}$
(14) s.t.
$\displaystyle\qquad\tilde{f}_{1}(\textbf{x})=\tilde{\beta}_{11}x_{1}x_{2}+\tilde{\beta}_{21}x_{2}x_{3}\leq
1,$ $\displaystyle\qquad x_{1},x_{2},x_{3}>0,$

where $\tilde{\beta}_{10}\sim\mathcal{LN}(A_{10},B_{10})$,
$\tilde{\beta}_{20}\sim\mathcal{LN}(A_{20},B_{20})$,
$\tilde{\beta}_{11}\sim\mathcal{LN}(A_{11},B_{11})$,
$\tilde{\beta}_{21}\sim\mathcal{LN}(A_{21},B_{21})$ are independent linear-
normal uncertain random variables, $A_{10}\sim\mathcal{N}(50,3)$,
$B_{10}\sim\mathcal{N}(40,2)$, $A_{20}\sim\mathcal{N}(45,2)$,
$B_{20}\sim\mathcal{N}(40,1)$, $A_{11}\sim\mathcal{N}(1,\frac{1}{3})$,
$B_{11}\sim\mathcal{N}(\frac{3}{2},\frac{2}{3})$,
$A_{21}\sim\mathcal{N}(\frac{2}{3},\frac{1}{3})$,
$B_{21}\sim\mathcal{N}(\frac{4}{3},1)$ are independent normal random
variables. Based on an $\alpha$ optimistic value point of view, the
deterministic form of Problem 14 with a tolerance level $\epsilon$ is

$\displaystyle\min\quad(40+10\alpha)x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}+(40+5\alpha)x_{1}x_{3}$
(15)
$\displaystyle\qquad+\Phi^{-1}(1-\epsilon)\sqrt{\big{(}9\alpha^{2}+4(1-\alpha)^{2}\big{)}x_{1}^{-2}x_{2}^{-2}x_{3}^{-2}+\big{(}4\alpha^{2}+(1-\alpha)^{2}\big{)}x_{1}^{2}x_{3}^{2}}$
s.t.
$\displaystyle\quad\frac{3-\alpha}{2}x_{1}x_{2}+\frac{4-2\alpha}{3}x_{2}x_{3}$
$\displaystyle\qquad+\Phi^{-1}(1-\epsilon)\sqrt{\bigg{(}\frac{1}{9}\alpha^{2}+\frac{4}{9}(1-\alpha)^{2}\bigg{)}x_{1}^{2}x_{2}^{2}+\bigg{(}\frac{1}{9}\alpha^{2}+(1-\alpha)^{2}\bigg{)}x_{2}^{2}x_{3}^{2}}\leq
1,$ $\displaystyle\quad
x_{1},x_{2},x_{3}>0,\alpha\in(0,1),\epsilon\in(0,0.5].$

To solve Problem 15, let us introduce two new variables, $x_{4}$ and $x_{5}$,
defined as
$x_{4}=\big{(}9\alpha^{2}+4(1-\alpha)^{2}\big{)}x_{1}^{-2}x_{2}^{-2}x_{3}^{-2}+\big{(}4\alpha^{2}+(1-\alpha)^{2}\big{)}x_{1}^{2}x_{3}^{2}$
and
$x_{5}=\big{(}\frac{1}{9}\alpha^{2}+\frac{4}{9}(1-\alpha)^{2}\big{)}x_{1}^{2}x_{2}^{2}+\big{(}\frac{1}{9}\alpha^{2}+(1-\alpha)^{2}\big{)}x_{2}^{2}x_{3}^{2}$.
Then, Problem 15 becomes as

$\displaystyle\min\quad(40+10\alpha)x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}+(40+5\alpha)x_{1}x_{3}+\Phi^{-1}(1-\epsilon)x_{4}^{\frac{1}{2}}$
(16) s.t.
$\displaystyle\quad\frac{3-\alpha}{2}x_{1}x_{2}+\frac{4-2\alpha}{3}x_{2}x_{3}+\Phi^{-1}(1-\epsilon)x_{5}^{\frac{1}{2}}\leq
1,$
$\displaystyle\quad\big{(}9\alpha^{2}+4(1-\alpha)^{2}\big{)}x_{1}^{-2}x_{2}^{-2}x_{3}^{-2}x_{4}^{-1}+\big{(}4\alpha^{2}+(1-\alpha)^{2}\big{)}x_{1}^{2}x_{3}^{2}x_{4}^{-1}\leq
1,$
$\displaystyle\quad\bigg{(}\frac{1}{9}\alpha^{2}+\frac{4}{9}(1-\alpha)^{2}\bigg{)}x_{1}^{2}x_{2}^{2}x_{5}^{-1}+\bigg{(}\frac{1}{9}\alpha^{2}+(1-\alpha)^{2}\bigg{)}x_{2}^{2}x_{3}^{2}x_{5}^{-1}\leq
1,$ $\displaystyle\quad
x_{1},x_{2},x_{3},x_{4},x_{5}>0,\alpha\in(0,1),\epsilon\in(0,0.5].$

In Problem 16, the total number of terms is $10$ and the total number of
variables is $5.$ So, the degree of difficulty is $=10-5-1=4.$ Let
$\delta_{1},\delta_{2},\ldots,\delta_{10}$ be the corresponding dual
variables. The the dual of Problem 16 is

$\displaystyle\max\quad
V(\delta)=\bigg{(}\frac{40+10\alpha}{{\delta_{1}}}\bigg{)}^{{\delta_{1}}}\bigg{(}\frac{40+5\alpha}{{\delta_{2}}}\bigg{)}^{{\delta_{2}}}\bigg{(}\frac{\Phi^{-1}(1-\epsilon)}{{\delta_{3}}}\bigg{)}^{{\delta_{3}}}\bigg{(}\frac{3-\alpha}{{2\delta_{4}}}\bigg{)}^{{\delta_{4}}}\bigg{(}\frac{4-2\alpha}{{3\delta_{5}}}\bigg{)}^{{\delta_{5}}}$
(17)
$\displaystyle\quad\bigg{(}\frac{\Phi^{-1}(1-\epsilon)}{{\delta_{6}}}\bigg{)}^{{\delta_{6}}}\bigg{(}\frac{9\alpha^{2}+4(1-\alpha)^{2}}{{\delta_{7}}}\bigg{)}^{{\delta_{7}}}\bigg{(}\frac{4\alpha^{2}+(1-\alpha)^{2}}{{\delta_{8}}}\bigg{)}^{{\delta_{8}}}\bigg{(}\frac{\alpha^{2}+4(1-\alpha)^{2}}{{9\delta_{9}}}\bigg{)}^{{\delta_{9}}}$
$\displaystyle\quad\bigg{(}\frac{\alpha^{2}+9(1-\alpha)^{2}}{{9\delta_{10}}}\bigg{)}^{{\delta_{10}}}(\delta_{4}+\delta_{5}+\delta_{6})^{\delta_{4}+\delta_{5}+\delta_{6}}(\delta_{7}+\delta_{8})^{\delta_{7}+\delta_{8}}(\delta_{9}+\delta_{10})^{\delta_{9}+\delta_{10}}$
s.t. $\displaystyle\qquad\delta_{1}+\delta_{2}+\delta_{3}=1,$
$\displaystyle\qquad-\delta_{1}+\delta_{2}+\delta_{4}-2\delta_{7}+2\delta_{8}+2\delta_{9}=0,$
$\displaystyle\qquad-\delta_{1}+\delta_{4}+\delta_{5}-2\delta_{7}+2\delta_{9}+2\delta_{10}=0,$
$\displaystyle\qquad-\delta_{1}+\delta_{2}+\delta_{5}-2\delta_{7}+2\delta_{8}+2\delta_{10}=0,$
$\displaystyle\qquad\frac{1}{2}\delta_{3}-\delta_{7}-\delta_{8}=0,$
$\displaystyle\qquad\frac{1}{2}\delta_{6}-\delta_{9}-\delta_{10}=0.$

Solving Problem 17 for a fixed $\alpha\in(0,1)$ with the tolerance level
$\epsilon=0.05$, we get the dual solution. Consequently, by the primal-dual
relationship, we find the primal solution. When we set parameter $\alpha$ to
discrete values from $0.1$ to $0.9$ with an increment $0.1$, the corresponding
solutions are computed in Table 1.

Table 1:  Optimal solutions  $\alpha$ | $x_{1}^{*}$ | $x_{2}^{*}$ | $x_{3}^{*}$ | $x_{4}^{*}$ | $x_{5}^{*}$ | Objective value
---|---|---|---|---|---|---
0.1 | 1.454 | 0.167 | 1.233 | 39.882 | 0.056 | 219.893
0.2 | 1.409 | 0.183 | 1.219 | 31.919 | 0.051 | 213.316
0.3 | 1.361 | 0.201 | 1.207 | 27.584 | 0.047 | 206.732
0.4 | 1.310 | 0.223 | 1.199 | 26.036 | 0.042 | 200.180
0.5 | 1.254 | 0.247 | 1.195 | 26.533 | 0.038 | 193.715
0.6 | 1.194 | 0.274 | 1.197 | 28.480 | 0.034 | 187.493
0.7 | 1.132 | 0.304 | 1.208 | 31.494 | 0.031 | 181.885
0.8 | 1.075 | 0.332 | 1.225 | 35.514 | 0.030 | 177.570
0.9 | 1.033 | 0.354 | 1.242 | 40.852 | 0.032 | 175.417

We find the deterministic form of Problem 14 based on an optimistic value
point of view. Further, we solve the transformed deterministic problem for
different values of $\alpha\in(0,1)$, which are given in Table 1. Similarly,
it is easy to find the solution to Problem 14 based on the pessimistic value
point of view and the expected value point of view. However, from Table 1, we
visualize the solution to Problem 14 based on the pessimistic value point of
view and the expected value point of view. Figure 6 shows the objective values
with respect to the parameter $\alpha$ based on the optimistic value point of
view and the pessimistic value point of view. However, $\alpha=0.5$ gives the
optimal objective value based on expected value point of view.

Figure 6: Optimal objective values with respect to the parameter $\alpha$.

##  6 Conclusion

In conclusion, our work delves into the realm of GP from a unique perspective
by incorporating independent linear-normal uncertain random variables into the
problem. We introduce and define the concept of an uncertain random variable
and propose a novel approach in the form of linear-normal uncertain random
variables. We are able to transform these uncertain random variables into
their stochastic counterparts by using three different criteria: optimistic
value, pessimistic value, and expected value. This turns the uncertain random
GP problem into a manageable stochastic GP problem.

One of the primary advantages of our approach is its ability to model and
address uncertainty in a structured manner, which is particularly useful in
real-world scenarios where data is not always deterministic. This method
allows decision-makers to make informed choices that consider the inherent
uncertainty and randomness in the input coefficients, ultimately leading to
more reliable solutions.

Furthermore, we present an equivalent deterministic representation of the
transformed GP problem, which simplifies the decision-making process for
practitioners who may prefer working with deterministic models. This duality
between stochastic and deterministic representations provides flexibility in
addressing the problem based on the decision-maker’s preference and data
availability.

However, it is essential to acknowledge the limitations of our work. Firstly,
the computational complexity of solving stochastic GP problems can be
substantial, especially for large-scale instances, and may require specialized
optimization techniques or approximation methods. Secondly, our approach
relies on the assumption that the coefficients are independent linear-normal
uncertain random variables, which might not always hold in practical
applications. Addressing correlated uncertainties remains an avenue for future
research. Additionally, the transformation of uncertain variables into
stochastic forms may introduce approximations, potentially impacting the
accuracy of the results.

In summary, our work offers a valuable contribution to the field of GP
problems by providing a systematic approach to handling uncertainty and
randomness through linear-normal uncertain random variables. While it offers
practical benefits in decision-making under uncertainty, decision-makers
should be aware of the computational challenges and the assumptions inherent
in our approach. Future research could explore more sophisticated techniques
to handle correlated uncertainties and further improve the applicability of
stochastic GP problems in real-world scenarios.

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 numerical example 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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# Translated Skip Connections - Expanding the Receptive Fields of Fully
Convolutional Neural Networks

###### Abstract

The effective receptive field of a fully convolutional neural network is an
important consideration when designing an architecture, as it defines the
portion of the input visible to each convolutional kernel. We propose a neural
network module, extending traditional skip connections, called the translated
skip connection. Translated skip connections geometrically increase the
receptive field of an architecture with negligible impact on both the size of
the parameter space and computational complexity. By embedding translated skip
connections into a benchmark architecture, we demonstrate that our module
matches or outperforms four other approaches to expanding the effective
receptive fields of fully convolutional neural networks. We confirm this
result across five contemporary image segmentation datasets from disparate
domains, including the detection of COVID-19 infection, segmentation of aerial
imagery, common object segmentation, and segmentation for self-driving cars.

Index Terms—  skip connections, receptive fields, fully convolutional neural
networks, semantic segmentation, dilated convolution

##  1 Introduction

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In this paper, we propose an extension of the skip connection module that
greatly increases the receptive fields of FCNs by taking advantage of their
encoder-decoder structure. As opposed to a traditional skip connection, which
skips an input from an earlier layer deeper into the network; we gradually
translate the skipped inputs spatially further from their centre as they
progress up the decoder. Since the decoder, composed of inverse convolutions,
continually increases the resolution of the feature maps as we translate
across the skipped input, the granularity of the translated receptive field
continually decreases the further we move from the original kernel.
Computational and memory burdens were an important consideration when
developing the proposed module, and we have kept it in-line with other
contemporary approaches to increasing receptive fields. We evaluate the
proposed module on five different segmentation domains, all of which vary in
resolution, data availability, and label complexity. The datasets range from
medical image segmentation for COVID-19 infection to segmentation for self-
driving cars. We aim to show that the increase in the receptive fields of FCNs
provided by the proposed module has the potential to improve performance on a
wide variety of image segmentation tasks.

Some background and an introduction to related work is required before we
describe the proposed module, we provide this in section 2. We then introduce
the proposed module, the translated skip connection (TSC), in section 3. In
section 4, we demonstrate the efficacy of the module empirically by embedding
it in an architecture and comparing it to other architectures that increase
the receptive field using strided pooling, dilated convolution, and strided
convolution. We end the paper in section 5 with concluding remarks and
describe some of the future work that could build upon the results shown here.

##  2 Background and Related Work

###  2.1 Receptive Fields

The “receptive field” of a pixel or unit in a layer of the network is defined
as the region of that layer’s input that non-negligibly impacts the unit in
question. This amounts to the region of the feature map covered by the kernel
that provided that unit’s value. If we propagate that concept back through an
entire network, we can find the region of the original input to the network
that affected the value of the unit under consideration. That region of the
original input is called the effective receptive field (ERF), or just the
“effective field” [1].

The ERF of the network often determines which tasks the network can undertake.
Tasks that are more reliant on the shape of objects than their texture, or
tasks that require the detection of very spatially dispersed objects, may be
impossible without effective fields large enough to include the relevant
features of the input space. There are a number of ways to increase the ERF of
a network: (1) increasing the depth of the network, (2) using larger
convolutional kernels, (3) resizing the input image, (4) using pooling, (5)
using strided convolution or strided pooling, and (6) using dilated
convolution.

###  2.2 Expanding Receptive Fields

Increasing the depth of the network, resizing the input image, using larger
convolutional kernels, and using pooling were all mentioned in section 2.1 as
methods that increase the receptive fields of convolutional neural networks.
All of these approaches have drawbacks; increasing the depth of neural
networks often leads to vanishing gradients and over-fitting, besides a
general increase in computational burden. Pooling and resizing the input image
can dramatically decrease the dimensionality of the input data and can result
in information loss, they also manipulate the data in a fixed way as opposed
to learnable techniques which are more adaptive. While using larger
convolutional kernels does increase the receptive field, it has been shown
that larger convolutional kernels are disproportionately computationally
expensive; architectures with more layers and smaller kernel sizes are more
efficient despite having the same number of parameters [2]. Strided
convolution provides a learnable way to increase the ERF of an FCN however it
typically replaces strided pooling, which means an increased number of
parameters and a greater propensity to overfit.

Dilated convolution [3] provides a way to maintain the kernel size while
increasing the receptive field of convolutional kernels in a memory efficient
way. We also define the dilation rate as the horizontal, vertical, or diagonal
distance from each weight to its nearest neighbours in the filter. Traditional
convolution is said to have a dilation rate of $1$. As shown by [4] and [5],
dilated convolution out-performs traditional convolution in a variety of
cases.

While dilated convolution does not require an increase in the number of
parameters used in each kernel, and so has similar memory requirements to
traditional convolution, it does create gaps in the kernel. These gaps mean
that the optimisations generally applied to matrix multiplication are not
effective when dilated convolution is being used; because of this, dilated
convolution often takes up to three times longer than traditional convolution
despite not using more memory.

###  2.3 Skip Connections

Skip connections [6] serve to counteract the problem of deeper networks
increasing training error through continually vanishing gradients. By
connecting layers in the network directly to the inputs of layers that
occurred earlier in the network, we curtail the tendency for weights and
biases to degrade the inputs of deeper layers. As weights deeper in the
network get smaller, these residual blocks may easily tend towards the
identity function instead of tending to an output of zero, particularly when
the ReLU activation function is used. Thus, skip connections allow us to add
layers to CNNs almost indiscriminately; if the layers are of no benefit then
the output of the network will not be altered.

We consider two different implementations of skip connections, additive and
concatenative. The first, $\textbf{f(x)}+\textbf{x}$, we refer to as an
“additive skip connection”. This is simply the element wise summation of the
output of one layer with the input of an earlier layer in the network. The
latter “concatenative skip connections”, denoted
$\textbf{f(x)}\oplus\textbf{x}$, concatenate the two layers together, and thus
requires extra parameters to process.

##  3 Translated Skip Connections

We propose translated skip connections (TSCs), a novel redesign of skip
connections that greatly increases the effective field of FCNs while only
marginally increasing the number of parameters in the network. The proposed
module also includes a traditional additive skip connection, thus still
avoiding the problem of vanishing gradients.

Fig. 1: Diagram of the proposed TSC architecture

We define the translation function $\textbf{T}:\mathbb{R}^{m\times
n}\to\mathbb{R}^{m\times n}$ that shifts an input matrix in a given direction
by a given factor. For example, given a feature map X,
$\textbf{T}_{\uparrow}(\textbf{X},\frac{1}{4})$ would translate the feature
map upwards one fourth the height of the feature map, with the top of the
feature map reentering at the bottom.

The TSC architecture used in our experiments is depicted in Fig. 1. The
architecture combines ideas from two earlier networks, called V-Net [7] and
U-Net [8]; we will refer to this architecture as TSC-Net. Each skip connection
- represented by the yellow arrows - includes a single additive skip
connection and three translated concatenative skip connections that translate
the encoder output left, up, and diagonally, respectively. The translation
factor in each TSC is $\frac{l}{\textbf{D}+1}$, where $l$ is the integer index
of the layer in the network, from $1$ to $5$ starting at the bottleneck, and D
is the network’s depth, $5$. Given the notation defined above, each of the
five skip connections in the network can be described with the following
equation, where X is the skipped input to the layer, l is the index of the
layer, and $\textbf{f}(\textbf{X})$ is the output of the previous layer:

$(\textbf{f}(\textbf{X})+\textbf{X})\oplus\textbf{T}_{\leftarrow}(\textbf{X},\frac{l}{6})\oplus\textbf{T}_{\uparrow}(\textbf{X},\frac{l}{6})\oplus\textbf{T}_{\nwarrow}(\textbf{X},\frac{l}{6})$
(1)

This equation comprises a single TSC, so each TSC is made of a single additive
skip connection, $\textbf{f}(\textbf{X})+\textbf{X}$, and three concatenative
skip connections, denoted $\oplus$.

The translational equivariance demonstrated by CNNs is typically considered
advantageous but it poses a problem for the proposed module. To ameliorate
this, we implement a variant of CoordConv [9] by concatenating the coordinate
of each pixel to its feature vector and normalising it. The transformation we
apply to each pixel’s feature vector is:

$(R,G,B)\rightarrow{(R,G,B,\frac{x}{W},\frac{y}{H})}$ (2)

yielding the five input channels of the network. Rather than translation
equivariance being inevitable, this modification makes it optional; we show
the benefit of this modification in the next section.

##  4 Experiments and Results

###  4.1 Methodology

The five architectures we consider are constructed to compare TSCs to the
other approaches of increasing the receptive field introduced in section 2.1.
Our implementations of these architectures, using Pytorch with Pytorch
Lightning, are available on GitHub under a GNU General Public
License222https://github.com/JoshuaDBruton/TSC.

The baseline model that we consider is the popular U-Net architecture [8].
U-Net uses $2\times 2$ strided max pooling to scale down the input image and
thus increase the effective field. We extend the U-Net architecture twice,
using dilated convolution with dilation rates of both two and three, as
dilation rates higher than three resulted in poor performance. The fourth
architecture we examine is a scaled down version of V-Net, which we refer to
as B-Net; the primary difference is replacing U-Net’s max-pooling with strided
convolution.

We compare the four architectures discussed above to the TSC architecture,
TSC-Net, depicted in Fig. 1. TSC-Net would have more parameters than U-Net and
B-Net, because of the two extra concatenative skip connections in each decoder
layer. To prove that it is the placement of TSC-Net’s parameters and not their
quantity that improves performance, we reduce the number of filters so that it
totals approximately $9.6$ million parameters, fewer than B-Net.

All five architectures are trained on five different datasets for 100 epochs;
they are: Landcover.ai [10], Inria Image Labelling [11], Carla [12], the
COVID-19 CT scans dataset [13, 14], and PASCAL VOC2012 [15]. We measure the
performance of all the architectures with the Mean Intersection Over Union
Metric [16], and report the average of the maximum validation MiOU attained by
each of the models.

###  4.2 Results

Fig. 2: Chart depicts the validation MIoU of four different models while
training on the top and left rectangle dataset, each line is averaged over
five training regimes, with the standard error depicted in a lighter shade

We must first establish whether TSC satisfies its intended purpose, that is,
geometrically expanding the receptive fields of FCNs. We also test whether a
network, after having its receptive field expanded by TSC, benefits from
optional translational equivariance (OTE). This experiment is conducted on a
synthetic texture analysis dataset available online
333https://webwtad.firebaseapp.com/. In this experiment, we make use of two
FCN architectures, U-Net and TSC-Net. We train these networks with and without
OTE on the synthetic dataset for 50 epochs each, averaging the results for
each network over five training runs. The results of this experiment are shown
in figure 2. As can be seen there, every addition of OTE and TSC results in a
performance improvement compared to the baseline model; but, more interesting
conclusions can also be drawn from this result.

Firstly, U-Net without OTE’s performance barely improves throughout training,
and - contrary to the norm - the stability of the model decreases the more it
trains, signifying its inability to solve this task. The rest of the
architectures all show convergence; however, the least stable architecture is
TSC-Net without OTE. We argue that this is because the information provided by
TSC is quite complex for the model to interpret without OTE. The model does
not know where in the image the kernels in the TSC portion of its feature map
are coming from. Two other results attest to this argument: when we add OTE to
TSC-Net, the stability of convergence notably improves, and, at the beginning
of training, U-Net with OTE performs better than TSC-Net without OTE until
TSC-Net learns to make use of the complex information the TSC module provides.

Table 1: Comparison of baseline and dilated convolution with TSC Average of
Maximum Mean Intersection Over Unionab
---
Dataset | U-Net | Dilation 2 | Dilation 3 | B-Net | TSC-Net
COVID-19 | 0.86 $\pm$ 8e-3 | 0.87 $\pm$ 7e-3 | 0.86 $\pm$ 5e-3 | 0.89 $\pm$ 2e-3 |  0.90 $\pm$ 1e-4
IILD | 0.72 $\pm$ 0.01 | 0.74 $\pm$ 3e-3 | 0.75 $\pm$ 4e-3 | 0.79 $\pm$ 2e-3 |  0.81 $\pm$ 1e-4
Landcover.ai | 0.45 $\pm$ 0.01 | 0.49 $\pm$ 6e-3 | 0.50 $\pm$ 9e-3 | 0.50 $\pm$ 5e-3 |  0.53 $\pm$ 6e-3
Carla | 0.73 $\pm$ 6e-3 | 0.71 $\pm$ 9e-3 | 0.70 $\pm$ 2e-3 |  0.76 $\pm$ 3e-3 |  0.76 $\pm$ 2e-3
VOC20212 | 0.48 $\pm$ 0.01 | 0.49 $\pm$ 0.01 |  0.50 $\pm$ 4e-3 | 0.44 $\pm$ 1e-3 |  0.50 $\pm$ 4e-3
aRanges from 0.0 to 1.0, closer to 1.0 is better
bAveraged over 3 trains for IILD and Landcover.ai, and 5 trains for the
remaining datasets

Table 1 contains experimental results for all five of the architectures we
considered on all five datasets. All architectures achieve their greatest MIoU
on the COVID-19 dataset, which is expected given that U-Net was initially
designed for application to medical imaging and the other architectures are
extensions of U-Net. B-Net’s strided convolution provides learnable down-
sampling and thus greater expressiveness, and in this domain did not cause
over-fitting because the training and testing sets are fairly consistent.
Despite not having strided convolution, TSC-Net’s translated skip connections
allow for a similar increase in expressiveness but also provide a larger ERF,
and so TSC-Net slightly outperforms B-Net.

Following the COVID-19 dataset, we present results from IILD and Landcover.ai,
two satellite imaging datasets with similar characteristics; unlike the
COVID-19 dataset, satellite image segmentation certainly requires large
receptive fields as objects typically span wide areas and their class is often
inferred from the surrounding region. IILD and Landcover.ai are also large
datasets, and so over-fitting is not a concern. This explains why larger, more
learnable, receptive fields consistently improve performance on both datasets.
Landcover.ai has a greater label-complexity, hence the lower MIoUs achieved by
all architectures on that dataset, however the relative performance of the
architectures was markedly similar in both domains.

The next result in Table 1 is for the CARLA segmentation for self-driving cars
dataset. It is a synthetic dataset, and so the textures and the gradients
between them are simplistic compared to real-world images; this necessarily
means that higher receptive fields will confer less benefit. Thus, the
expressivity of each architecture will have more impact on their performance.
B-Net’s learnable down-scaling and TSC-Net’s learnable skip connections are
the source of their extra expressivity and better performance.

Finally, we present the results for the PASCAL VOC2012 dataset; the most
complex domain we consider, it contains 22 labels, so random performance would
result in an MIoU below 0.05. VOC2012 is also a small dataset, with
approximately 1,500 training images, adding to the difficulty. Typically,
researchers would apply deeper architectures to solve this domain, but since
our primary concern is relative performance, our models are sufficient. For
the first time, B-Net achieved the worst performance of all the architectures,
this is likely due to overfitting. Only two of the architectures achieved the
satisfactory margin of 0.5, the dilated model with dilation rate 3, and TSC-
Net. Unsurprisingly, these are the architectures with the largest, learnable
receptive fields; these receptive fields are an important quality on this
dataset due to the large amount of correlated classes, for example, horse and
grass or desk and monitor. Both of these architectures also make use of
strided pooling when down-sampling, this counteracts the overfitting that
befell B-Net’s strided convolution.

##  5 Conclusion

In this paper, we proposed a module for expanding the receptive fields of
FCNs. We also investigated alternative approaches to increasing the receptive
fields of FCNs. We considered dilated convolution, strided pooling, strided
convolution, and, our proposed module, TSCs. Our experimental results,
acquired across five disparate image segmentation datasets, show that using
TSCs led to competitive or superior performance when compared to the other
methods considered in this paper.

Future work should consider alternative implementations of TSCs that could
extend to traditional neural networks. Alternatively, conducting more
experiments with TSCs embedded in larger architectures that combine the
techniques compared in this manuscript would also be beneficial. Finally, our
results suggest that including strided pooling and strided convolution in an
alternating fashion in the encoder layer of an FCN could simultaneously
improve expressivity and curb over-fitting, which could provide for an
interesting finding.

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