Comparative Study of R Programming and Lingo Software in Fully
Fuzzy Linear Programming Problems
MOHAMED SALIH MUKTHAR M1,a,*, RAMATHILAGAM S 2,b
1Department of Mathematics, C. Abdul Hakeem College (Autonomous),
(Affiliated to Thiruvalluvar University, Serkkadu, Vellore – 632115),
Melvisharam-632509, INDIA
2Department of Mathematics, Periyar Arts College,
(Affiliated to Thiruvalluvar University, Serkkadu, Vellore – 632115),
Cuddalore-607001, INDIA
* Correspondence: salih.mat@cahc.edu.in
aOrcid ID: https://orcid.org/0000-0002-2457-5314
bOrcid ID: https://orcid.org/0009-0002-4145-5629
Abstract: The paper begins with an introduction to the concept of fuzzy optimization, providing an in-depth
explanation of its significance and applications. It then proceeds to define two distinct ranking functions that is
Yager’s and – ranking functions within the context of fuzzy optimization along with the properties of -
ranking function. Following this, a comprehensive overview of Lingo software and R programming is presented,
delving into their respective features and capabilities. This paper further delves into a detailed comparison of
Lingo and R programming, offering a thorough exploration of their strengths and limitations through the use of
example problems. These examples illustrate the process of computing optimal solutions for fully fuzzy linear
programming problems using both Lingo and R programming, providing a comprehensive and practical
comparison of their effectiveness in this context.
Key-words: Fuzzy optimization, Lingo, R programming, Fully fuzzy linear programming
Received: March 7, 2024. Revised: August 13, 2024. Accepted: September 11, 2024. Published: October 14, 2024.
1. Introduction
Optimization is the process of finding
the best solution, often involving minimizing or
maximizing the value of the objective.
Optimization problems revolve around
maximizing or minimizing a function within a
given set, typically representing a range of
available choices in a specific situation. This
function allows for the comparison of different
choices to determine the best option. The fuzzy
concept addresses issues related to ambiguous,
subjective, and imprecise judgments. It
quantifies the linguistic aspect of available data
and preferences for individual or group
decision-making.
The concept of solving fuzzy
optimization, particularly fuzzy linear
programming problems, has garnered attention
in the academic community since the first
works in fuzzy optimization were published in
1974. Fuzzy Linear Programming model was
introduced by Zimmermann, who proposed a
method to solve Linear Programming problems
with fuzzy linear constraints. His contributions
have influenced many other works and opened
the door to involve fuzzy reasoning in
optimization. Therefore, it is important to refer
to related fuzzy mathematical models and
methods. Numerous related works, extensions,
and applications have been published after
Zimmermann's seminal works in 1977. Various
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scholars, including mathematicians,
economists, and engineers, have contributed to
the development of fuzzy mathematical
programming based on the principles given by
Bellman and Zadeh, who provided a unified
framework for fuzzy decision-making. More
specific reviews were published by
Ebrahimnejad and Verdegay [1], [2] who
reviewed several solution methods for FLPs
according to the classification proposed by
Shams et al. [3], who published a brief review
on several FLP models and suggested three
classes only; Ghanbari et al. [4], who recently
published an extensive review of methods for
solving FLPs and classified them into five
categories depending on fuzzy parameters; and
Sotoudeh-Anvari [5], who has made an
interesting criticism about some drawbacks and
mathematical incorrect assumptions in fuzzy
OR methods, including fuzzy linear
programming problems from 2010 to 2020.
2. Materials
In this section, first, some basic concepts
will be discussed.
Definition 2.1
A fuzzy number
= is said to
be a trapezoidal fuzzy number [6] if its membership
function is given by:
Definition 2.2
A trapezoidal fuzzy number
=
( can be shown by support and core
as in the Figure -1.
Its support is []
and core is [].
1
Figure-1
Definition 2.3
A fuzzy linear programming problem in
which it contains all the parameters are fuzzy is
called a fully fuzzy linear programming
problem.
It can be defined as follows [6]:
Maximize (or Minimize)
Subject to the constraints
, i = 1, 2, 3, …, m
, for all j,
where
,
,
,
are trapezoidal fuzzy
numbers.
Definition 2.4
Yager’s ranking function [7] of a
trapezoidal fuzzy number
= (
is defined as
(
) =
Definition 2.5
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– ranking function of a trapezoidal
fuzzy number
= is defined as
(
) =
. This value have
found by Mary George and Savitha M.T [8]
using the idea of Adrian.I Ban and Lucian
Coroianu [9].
Proposition 2.5.1
For any trapezoidal fuzzy numbers
and
,
Proof
Let
= and
=
By definition, (
) =
(
) = (
)
(
)+ (
)
Proposition 2.5.2
For any finite trapezoidal fuzzy
numbers
,
…,
Proof is consequence of proposition 2.5.1.
Proposition 2.5.3
For any trapezoidal fuzzy number
,
, where is a scalar.
Proof
Let
=
By definition, (
) =
3. Overview of Lingo and R
Programming
The LINGO software [10] is a powerful
tool specifically designed to effectively handle
a wide range of optimization problems. It is
capable of addressing linear, non-linear, mixed
integer linear, and non-linear programming
challenges. Within LINGO, the Solver
command is utilized to find solutions to these
optimization problems, employing three
distinct algorithms: Simplex, Dual Simplex,
and Barrier.
It is crucial to understand that LINGO
is adept at solving crisp linear programming
problems but does not offer direct support for
solving fuzzy linear programming problems.
However, it is possible to leverage LINGO to
obtain optimal solutions for fuzzy linear
programming problems by first converting
them into crisp linear programming problems.
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R is a programming language [11]
primarily used for statistical computation,
graphical representation, visualization, and
data analysis. It runs on multiple platforms,
such as Windows, Mac, Linux, etc.
Additionally, R is an open-source software and
it has an extensive R archive network that
contains more than 100000 packages.
The techniques used to solve fully fuzzy
linear programming problems vary depending
on the specific nature of the problem. The fuzzy
algebraic method and fuzzy simplex method
are two prominent approaches employed in
certain cases. However, it is important to
acknowledge that no single method can
comprehensively address all types of fully
fuzzy linear programming problems. In
addition to these two methods, there are various
other specialized techniques available for
addressing specific scenarios. Currently, R
programming is increasingly being used to
solve most types of fully fuzzy linear
programming problems.
3.1 R Commands for Fuzzy Linear
Programming
The following are the R commands [12]
for applying fuzzy linear programming
problems.
FuzzyLP
It is a package to solve fuzzy linear
programming problems in R.
FCLP.classicObjective
It is used to find the solution of the
linear programming problem which contains
fuzzy constraints and crisp objective.
FCLP.fixedBeta
It solves a fuzzy linear programming
problem with fuzzy constraints.
FOLP.multiObj
It solves a fuzzy objective linear
programming problem using the
Representation Theorem through a
multiobjective approach.
FOLP.posib
It solves a fuzzy objective linear
programming problem using the
Representation Theorem through a possibilistic
approach.
FOLP.ordFun
It solves a fuzzy objective linear
programming problem using ordering
functions.
GFLP
It is used to solve fully fuzzy linear
programming problems that have fuzzy
coefficients in the constraints, the objective
function and the technological matrix.
FuzzyNumbers
It is an open source (LGPL 3) package
for R. It provides S4 classes and methods to
deal with fuzzy numbers.
TrapezoidalFuzzyNumber()
It represents trapezoidal fuzzy numbers
in R.
4. Problem Statements and
Solutions
Problem Statement 4.1
A software company produces two
types of system-level solutions. It has been
estimated that the production of a system-level
solution of type 1 needs around 3 working
months (w.m.) for the construction of the
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knowledge base, about 3.5 w.m. for the
selection of the suitable reasoning and
inference methodology, and nearly 1 w.m. for
the choice of the proper intelligent authoring
shells. Also, the production of a system-level
solution of type 2 needs around 1, 3, and 2 w.m.
respectively, for each of the above procedures.
According to the company’s existing number of
specialized staff, at most 20 w.m. per year can
be spent for the construction of the knowledge
base, at most 30 w.m. for the selection of the
reasoning and inference methodology, and at
most 18 w.m. for the selection of the intelligent
authoring shells. If the net profit from the sale
of a system-level solution of type 1 nears 300
thousand euros and of a system-level solution
of type 2 nears 400 thousand euros, find how
many system-level solutions of type 1 and type
2 should be produced per year to maximize the
company’s total profit.
Solution
This problem is having uncertainty at
various parameters. So, using trapezoidal fuzzy
numbers, it can be formulated into a fully fuzzy
linear programming problem.
Let q1 and q2 represent the quantities of
system-level solutions of type 1 and type 2
respectively to be produced per year.
Then, the problem is mathematically
formulated as follows:
Max Z = (2.6, 2.8, 3.0, 3.2) q1 + (3.6, 3.8, 4.0, 4.2) q2
Subject to the constraints
(2.6, 2.8, 3.0, 3.2) q1 + (0.6, 0.8, 1.0, 1.2) q2 (19.6, 19.8, 20.0, 20.2)
(3.1, 3.3, 3.5, 3.7) q1 + (2.6, 2.8, 3.0, 3.2) q2 (29.6, 29.8, 30.0, 30.2)
(0.6, 0.8, 1.0, 1.2) q1 + (1.6, 1.8, 2.0, 2.2) q2 (17.6, 17.8, 18.0, 18.2)
q1, q2 0
To use LINGO software for this problem, first convert it into a crisp linear programming
problem by using – ranking function (
) =
.
Corresponding Crisp LPP is
Max Z = 2.9 q1 + 3.9 q2
Subject to the constraints
2.9 q1 + 0.9 q2 19.9
3.4 q1 + 2.9 q2 29.9
0.9 q1 + 1.9 q2 17.9
q1, q2 0
Applying Lingo software, the output will be as follows:
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Global optimal solution found.
Objective value: 38.08182
Infeasibilities: 0.000000
Total solver iterations: 2
Elapsed runtime seconds: 0.71
Model Class: LP
Total variables: 2
Nonlinear variables: 0
Integer variables: 0
Total constraints: 4
Nonlinear constraints: 0
Total nonzeros: 8
Nonlinear nonzeros: 0
Variable Value Reduced Cost
Q1 1.272727 0.000000
Q2 8.818182 0.000000
Row Slack or Surplus Dual Price
1 38.08182 1.000000
2 8.272727 0.000000
3 0.000000 0.5194805
4 0.000000 1.259740
Hence the optimal solution has been achieved with quantities of system-level solution type 1
at 1.272727 and type 2 at 8.818182 to maximize company profit of 38%.
Next, the same problem will be solved by using the R program.
For this, the following are the R commands to enter into the software:
> obj=c(FuzzyNumbers::TrapezoidalFuzzyNumber(2.6,2.8,3,3.2),FuzzyNumbers::TrapezoidalFuzzyNumber(3.6,3.8,4,4.
2))
> a11=FuzzyNumbers::TrapezoidalFuzzyNumber(2.6,2.8,3,3.2)
> a12=FuzzyNumbers::TrapezoidalFuzzyNumber(0.6,0.8,1,1.2)
> a21=FuzzyNumbers::TrapezoidalFuzzyNumber(3.1,3.3,3.5,3.7)
> a22=FuzzyNumbers::TrapezoidalFuzzyNumber(2.6,2.8,3,3.2)
> a31=FuzzyNumbers::TrapezoidalFuzzyNumber(0.6,0.8,1,1.2)
> a32=FuzzyNumbers::TrapezoidalFuzzyNumber(1.6,1.8,2,2.2)
> A=matrix(c(a11,a21,a31,a12,a22,a32),nrow = 3)
> dir=c("<=","<=","<=")
> b=c(FuzzyNumbers::TrapezoidalFuzzyNumber(19.6,19.8,20,20.2),FuzzyNumbers::TrapezoidalFuzzyNumber(29.6,29.
8,30,30.2),FuzzyNumbers::TrapezoidalFuzzyNumber(17.6,17.8,18,18.2))
> t=c(FuzzyNumbers::TrapezoidalFuzzyNumber(1,1,1,1),FuzzyNumbers::TrapezoidalFuzzyNumber(1,1,1,1),FuzzyNum
bers::TrapezoidalFuzzyNumber(1,1,1,1))
> sol=FuzzyLP::GFLP(obj,A,dir,b,t,maximum = TRUE,ordf_obj = "Yager1",ordf_res = "Yager3")
> sol
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Press the enter key to obtain the following output.
beta x1 x2
[1,] 0 1.012987 9.467532
[2,] 0.25 1.077922 9.305195
[3,] 0.5 1.142857 9.142857
[4,] 0.75 1.207792 8.980519
[5,] 1 1.272727 8.818182
sol[,"objective"]
[[1]]
Trapezoidal fuzzy number with:
support=[36.7169,43.0052],
core=[38.813,40.9091].
[[2]]
Trapezoidal fuzzy number with:
support=[36.3013,42.5312],
core=[38.3779,40.4545].
[[3]]
Trapezoidal fuzzy number with:
support=[35.8857,42.0571],
core=[37.9429,40].
[[4]]
Trapezoidal fuzzy number with:
support=[35.4701,41.5831],
core=[37.5078,39.5455].
[[5]]
Trapezoidal fuzzy number with:
support=[35.0545,41.1091],
core=[37.0727,39.0909].
Optimum solution
z = (37.0727, 39.0909, 2.0812, 2.0812)
For defuzzification, use the following Yager’s ranking function
(
) =
Therefore, Z = 37.9431.
From the R program, the output is Z = 37.9469 when q1 = 1.272727 and q2 = 8.818182.
Similar optimal solutions have been obtained with the same quantities of system-level solution
type 1 at 1.272727 and type 2 at 8.818182 in order to maximize the company's profit by 38%.
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Problem Statement 4.2
A social network under construction is
planning to use three types of hardware,
denoted as F1, F2, and F3. The cost of each unit
varies between 3800 - 4200 euros for F1, 1700
- 2300 euros for F2, and 5500 - 6500 euros for
F3. It has been estimated that each unit of F1 has
the capacity to satisfy the needs of about 150 -
250 male visitors and 400 - 600 female visitors
per day. Each unit of F2 can accommodate
about 320 - 480 male visitors and 60 – 140
female visitors, while each unit of F3 can cater
to about 170 - 230 male visitors and 80 - 120
female visitors per day. The network is
expected to have at least 2400 male and 800
female visitors per day. Determine the number
of units of each type that should be ordered to
minimize the cost of the hardware.
Solution
In this problem, each unit cost varies.
Types of hardware and network capacities are
uncertain.
So, using trapezoidal fuzzy numbers, it
can be formulated into a fully fuzzy linear
programming problem as follows:
Let h1, h2 and h3 respectively be the units of the
hardware F1, F2 and F3 that should be
ordered.
Then the problem is mathematically formulated
as follows:
Min z = (36, 38, 40, 42) h1 + (17, 19, 21, 23) h2 + (56, 59, 62, 65) h3
Subject to the constraints
(1.5, 1.8, 2.1, 2.4) h1 + (3.2, 3.7, 4.2, 4.7) h2 + (1.7, 1.9, 2.1, 2.3) h3 (23.0, 23.5, 24.0, 24.5)
(4.0, 4.7, 5.4, 6.1) h1 + (0.6, 0.9, 1.2, 1.5) h2 + (0.8, 1.0, 1.2, 1.4) h3 (7.0, 7.5, 8.0, 8.5)
h1, h2, h3 0
To use LINGO software, first convert this problem into a crisp linear programming problem
by using – ranking function (
) =
as follows:
Min z= 39 h1 + 20 h2 + 60.5 h3
Subject to the constraints
1.95 h1 + 3.95 h2 + 2 h3 23.75
5.05 h1 + 1.05 h2 + 1.1 h3 7.75
h1, h2, h3 0
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When using Lingo software, the output will be as follows:
Global optimal solution found.
Objective value: 129.4874
Infeasibilities: 0.000000
Total solver iterations: 2
Elapsed runtime seconds: 0.04
Model Class: LP
Total variables: 3
Nonlinear variables: 0
Integer variables: 0
Total constraints: 3
Nonlinear constraints: 0
Total nonzeros: 9
Nonlinear nonzeros: 0
Variable Value Reduced Cost
H1 0.3170391 0.000000
H2 5.856145 0.000000
H3 0.000000 46.72039
Row Slack or Surplus Dual Price
1 129.4874 -1.000000
2 0.000000 -3.354749
3 0.000000 -6.427374
Thus, the optimum solution is h1 = 0.3170391, h2 = 5.856145, h3 = 0 in order to minimize the
cost of hardware by 129.4874.
Next, this problem will be solved by using the R program.
The following R commands are to be used for finding the optimum solution.
> obj=c(FuzzyNumbers::TrapezoidalFuzzyNumber(36,38,40,42),FuzzyNumbers::TrapezoidalFuzzyNumber(17,19,21,23
),FuzzyNumbers::TrapezoidalFuzzyNumber(56,59,62,65))
> a11=FuzzyNumbers::TrapezoidalFuzzyNumber(1.5,1.8,2.1,2.4)
> a12=FuzzyNumbers::TrapezoidalFuzzyNumber(3.2,3.7,4.2,4.7)
> a13=FuzzyNumbers::TrapezoidalFuzzyNumber(1.7,1.9,2.1,2.3)
> a21=FuzzyNumbers::TrapezoidalFuzzyNumber(4,4.7,5.4,6.1)
> a22=FuzzyNumbers::TrapezoidalFuzzyNumber(0.6,0.9,1.2,1.5)
> a23=FuzzyNumbers::TrapezoidalFuzzyNumber(0.8,1,1.2,1.4)
> A=matrix(c(a11,a21,a12,a22,a13,a23),nrow = 2)
> dir=c(">=",">=")
> b=c(FuzzyNumbers::TrapezoidalFuzzyNumber(23,23.5,24,24.5),FuzzyNumbers::TrapezoidalFuzzyNumber(7,7.5,8,8.5
))
> t=c(FuzzyNumbers::TrapezoidalFuzzyNumber(0,0,0,0),FuzzyNumbers::TrapezoidalFuzzyNumber(0,0,0,0))
> sol=FuzzyLP::GFLP(obj,A,dir,b,t,maximum = FALSE,ordf_obj = "Yager1",ordf_res = "Yager3")
> sol
Press the enter key to obtain the following output.
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beta x1 x2 x3
[1,] 0 0.1550279 5.682961 0
[2,] 0.25 0.1955307 5.726257 0
[3,] 0.5 0.2360335 5.769553 0
[4,] 0.75 0.2765363 5.812849 0
[5,] 1 0.3170391 5.856145 0
> sol[,"objective"]
[[1]]
Trapezoidal fuzzy number with:
support=[102.191,137.219],
core=[113.867,125.543].
[[2]]
Trapezoidal fuzzy number with:
support=[104.385,139.916],
core=[116.229,128.073].
[[3]]
Trapezoidal fuzzy number with:
support=[106.58,142.613],
core=[118.591,130.602].
[[4]]
Trapezoidal fuzzy number with:
support=[108.774,145.31],
core=[120.953,133.131].
[[5]]
Trapezoidal fuzzy number with:
support=[110.968,148.007],
core=[123.314,135.661].
Optimum solution is
z = (123.314, 135.661, 12.346, 12.346)
For defuzzification, use the following Yager’s ranking function
(
) =
Hence, Z = 128.6644
Thus, the hardware cost reduction of 128.6644 will occur when ordering F1 is 0.3170391, F2 is
5.856145 and F3 is 0.
5. Discussion
In comparison, the R programming
language can be directly used to solve fully
fuzzy linear programming problems, while the
Lingo software requires these problems to be
converted into crisp linear
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programming problems before they can be
solved. In both models (problems 4.1 and 4.2),
it has been observed that the optimal solutions
obtained using Lingo software and R
programming are approximately similar.
6. Conclusion
In the initial sections of the paper, the
discussion centered around the significance of
fuzzy optimization concepts and their practical
applications. This involved a detailed
exploration of two distinct ranking functions,
along with a comprehensive explanation of
their respective properties. Furthermore, the
paper delved into an in-depth analysis of the
features associated with Lingo software and R
programming.
Following this, the Lingo software was
leveraged to identify the optimal solutions for
sample models of fully fuzzy linear
programming problems. This process involved
the conversion of these models into crisp linear
programming problems, culminating in the
successful determination of optimal solutions.
Similarly, the R programming language was
employed to derive optimal solutions for the
same models.
Subsequently, a comparative analysis
was conducted to juxtapose the outcomes
obtained through both methodologies, thereby
providing a comprehensive assessment of their
respective efficacy and applicability.
Acknowledgement:
The authors would like to thank the
editorial team for useful suggestions and feedback.
Contribution of Individual Authors to the Creation
of a Scientific Article:
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Report potential sources of funding:
No funding was received for conducting this
study.
Conflict of Interest:
The authors have no conflicts of interest to
declare that are relevant to the content of this article.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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_US
Engineering World
DOI:10.37394/232025.2024.6.11
Mohamed Salih Mukthar M., Ramathilagam S.
E-ISSN: 2692-5079
117
Volume 6, 2024
