Efficient Ranking Function Methods for Fully Fuzzy Linear Fractional
Programming problems via Life Problems
REBAZ MUSTAFA, NEJMADDIN SULAIMAN
Department of Mathematics,
College of Education,
Salahaddin University,
Erbil, IRAQ
Abstract: - In this paper, we propose two new ranking function algorithms to solve fully fuzzy linear fractional
programming (FFLFP) problems, where the coefficients of the objective function and constraints are
considered to be triangular fuzzy numbers (TrFN) s. The notion of a ranking function is an efficient approach
when you want to work on TrFNs. The fuzzy values are converted to crisp values by using the suggested
ranking function procedure. Charnes and Cooper’s method transforms linear fractional programming (LFP)
problems into linear programming (LP) problems. The suggested ranking functions methods' applicability to
actual problems of daily life, which take data from a company as an example, is shown, and it presents
decision-making and exact error with the main value problem. The study aims to find an efficient solution to
the FFLFP problem, and the simplex method is employed to determine the efficient optimal solution to the
original FFLFP problem.
Key-Words:- Linear Fractional Programming, Fully Fuzzy Linear Fractional Programming, Linear
Programming Problem, Ranking Function, Triangular Fuzzy Number, Charnes Cooper’s Method.
Received: August 29, 2021. Revised: August 11, 2022. Accepted: September 15, 2022. Published: October 10, 2022.
1 Introduction
A particular class of non-linear programming
problems is called linear fractional programming
(LFP) problems, in which the constraints are linear
equations (inequality) and the objective function is a
ratio of linear functions. The significance of this
class of problems becomes apparent as there are
many situations in corporate, economic, industrial,
etc., where it has to be achieved that the ratio of two
values, such as input/output, profit/cost,
nurse/patient, sales/stock, etc., has to be optimized.
Many academics, including Charnes Cooper [1],
Dinkelbach [2], Swarup [3], Guzel [4], Mustafa and
Sulaiman [5], [6], and Nawkhass and Sulaiman [7],
have developed a variety of ways to solve fractional
programming and multi-objective linear fractional
programming.
The fuzzy optimization problem is interesting and
finds use in many crucial areas [8], [9]. In real-
world situations, we frequently face the necessity to
optimize an objective function. Due to this, the
problem of linear fractional programming is
extended to include fully fuzzy programming
problems. In the literature, several approaches have
been put forward to solve fuzzy programming
problems.
Researchers have developed many strategies to find
exact, close, rough, or efficient solutions to the
fuzzy programming (FP) problem, as introduced by
the first technique of Nayak [10] and the approach
proposed by Umadevi [11]. Then, the researchers
used the ranking function to change the FP problem
into an appropriate crisp programming (CP)
problem; this is a novel approach to solving FP
problems.
A new viewpoint on how to solve fuzzy
programming problems is presented by the
application of fuzzy set theory [12] to deal with
optimization problems. There are other approaches,
including linguistic techniques based on Zadeh's
work [13] fuzzy goal programming approach [14],
Assignment Problem using Genetic Approach [15],
etc...
Likewise, after them, the ranking function is used to
explain the fuzzy fractional programming (FFP)
problems [16], multi-objective LFP problems with
fuzzy parameters [17], fully fuzzy multi-objective
LP problems [18], and fuzzy LP problems [19],
[20].
Many ranking function methods exist, such as the
area between the centroid and original point method
[21], SD of the PILOT procedure [22], the area
method [23]and a revised approach to the PILOT
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ranking procedure [24], and new ranking procedures
[19], new ranking [25], comparing different
rankings [26], ranking method [27] efficient
algorithm [16], Rouben ranking function [28], etc.
To deal with such kinds of inexact states, LFP
problems with fuzzy coefficients were introduced,
which are mentioned as FFLFP problems. The
FFLFP problem is a fantastic tool for making
decisions. With one or more objective functions,
such as profit/cost, inventory/sales,
output/employees, etc., it is used to model actual
problems. The FFLFP problem might be used
effectively since the procedure of production
planning involves increasing profit by lowering
costs with ambiguous values.
Applications of this problem include supply chain
management, network banning, traffic, and
transportation process of assignment, security, and
injury strategic planning, policy decisions, power
management, tax problems, and other actual
problems.
In the field of implementation in the life of society,
scholars have taken many serious steps, including
Garrido et al. [26] comparing different ranking
functions for solving fuzzy linear programming
problems with fuzzy cost coefficients from the field
of tourism. Mitlif [20] used an efficient ranking
function method to solve LP problems with
trapezoidal fuzzy coefficients in food products.
Stanojevic et al. [29] and Sapan et al. [30] proposed
a new approach for solving FFLFP problems and
provided a tool for making good decisions in
production planning separately. Also, Sapan [31]
presented a new formulation of the FFLFP problem
in a real-life case and found an efficient solution.
Regarding COVID-19, Sapan and Chakraborty [32]
proposed a new method to solve the LP problem
under a pentagonal fuzzy environment. Veeramani
and Sumathi [33], [34] studied a new approach to
solving the LFP problem with triangular fuzzy
coefficients about company manufacturing. Many
other researchers are constantly trying to do new
things for the benefit of society through fuzzy
number solutions.
Optimization problems have more formulations of
the objective function, such as linear programming,
quadratic programming, and fractional
programming. The coefficients of problems in the
objective function, constraints coefficient, or
decision-making variable are sometimes fuzzy
numbers. In this work, we focus on the FFLFP
problem under TrFNs. Here, we have received an
example from companies working in Erbil to
enforce our techniques and show the optimal results
for making the best decisions as well as to make
clear the company's owner in its profits and costs.
Here, we provide two model ranking function
techniques for dealing with the FFLFP problem and
de-fuzzing any fuzzy number. By taking advantage
of Charnes and Cooper’s method, the LFP problem
is converted into an LP problem. Also, by using the
simplex method, we obtain an optimal solution. The
derivations are used to clarify the techniques that are
being taught.
.
This article is constructed into six sections. In
section 2, some important preliminaries on
triangular fuzzy numbers and the notions that
benefit us from this work are presented. In section 3,
we show the mathematical problem formulation.
The ranking functions are derived in section 4. An
algorithm is illustrated in section 5. A real-life
example of an FFLFP problem is included in section
6. In section 7, we describe the discussion. Finally,
conclusions are described in section 8.
2 Preliminaries Notions
The fundamental principles of TrFNs with their
operations and some concepts are given in this
section.
Definition 1 Let is a universal set. Then the
ordered pairs of is a
subset in called fuzzy set, Where is
named membership function.
Definition 2 If is a fuzzy subset convex, normal,
and possesses bounded support of the set R is a real
number, then is said to be a fuzzy number.
Definition 3 A fuzzy number is called TrFN if its
representation is in the form with
, and the membership function is given by
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Fig. 1: TrFN Membership Function.
Definition 4 Let form and form
be two TrFNs,
where. Then the arithmetic
operations and scalar multiplications are defined by
1-
2-
3- where and
where
4-
Definition 5 A TrFN is called non-
negative (non-positive) TrFN iff (
Definition 6 D is a crisp set and it is also called a
classical set defined as the group of elements
present over the universal set U., in this case, a
random element is present that may be a part of D or
not which means two ways are possible to define the
set. These are the first element that would become
from set D, or it does not come from D, Crisp set
defines the value as either 0 or 1.
Definition 7 The crisp set of elements that belong to
at least to degree is named as - cut set
Definition 8 Defuzzification is the process of
changing fuzzy parameters into clear (crisp)
parameters. The set of all fuzzy sets can be mapped
onto the set of all real numbers using a technique.
Theorem 1 A feasible point is said to be an
optimal solution to any optimization problem,
if there does not exist such that
. Where
Remark 1 is said to be an efficient solution to a
linear programming problem if is feasible and no
other solution exists such that and
to maximize the problem and
to minimize the problem.
3 Mathematical Statement Problems
In this section, we will focus on the formulas we
worked on for this paper and used to express our life
problems.
3.1 Linear Fractional Programming (LFP)
Problem:
Subject to:
Where is an real
matrix and is a real number.
3.2 Fully Fuzzy Linear Fractional
Programming (FFLFP) problems:
In the FFLFP problem, the variables, and
coefficients are completely TrFNs.
The general form of the FFLFP problem is as
follows:
Subject to:
Where 1 is by n matrices, is n by 1 matrix,
is
m by n matrix, and is m by 1 matrix. Here all the
parameters
, and
,TrFNs
1
1
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4 Ranking Functions Derivation
A ranking function of fuzzy number,
is the set of all fuzzy numbers defined on
which maps each fuzzy number into a real number.
Here we use two different rankings.
4.1 First Ranking Function
A triangular fuzzy number defines on
the x-axis the real points we can divide the
entire range into six
intervals.
In the beginning, find the arithmetic mean of each
interval, the ranking function is defined by the
quadratic mean of an average mean of all such
possible intervals. Therefore, the ranking function
Note that, regarding being real
numbers in the above ranking function, we have
seven cases:
1- If are positive real numbers.
2- If is negative, and is positive.
3- If and are positive.
4- If is negative and are positive.
For the above situations, the ranking function is
5- If are negative real numbers then
the ranking function is
6-
7- If are negative real numbers and
then
8-
9- If are negative and is positive the
ranking function is
10-
4.2 Second Ranking Function
Several approaches for the ranking of fuzzy
numbers have been proposed in the past decade. An
efficient approach for comparing the fuzzy numbers
is by the use of a ranking function based on their
graded means.
Now, by using the following triangular membership:
Then, by using cut, where and
and ,
or
Where, is the lower bound and
is the
upper bound, then presented for arbitrary TrFNs be
an ordered pair of function [
, where
, suppose that is a weight for
and is a weight for
where .
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4.3 Note
If and are two fuzzy numbers then
1-
2-
3-
4-
5-
6-
5 Algorithm
In this algorithm, we describe the steps of the
solution to specific problems and similar problems
in our problem statement for this work.
1- The LFP problem with fixed coefficients is
converted into an LP problem by charnes
cooper’s method.
2- To obtain the optimal solution, solve the LP
problem using the simplex method.
3- Convert the LFP problem to an FFLFP
problem (by the values of the problem)
4- Convert the FFLFP problem to an LFP
problem, i.e., convert each fuzzy number
involved in the FFLFP problem to a crisp
number using ().
5- Solve the LFP problem by repeating steps
1-2. This solution is an efficient solution
when an optimal solution is required.
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6 Implementation Ranking Functions
in Life Problem
One of Erbil's well-known companies produces teak
wood household furniture for governmental and
non-governmental institutions, such as cupboards,
desks, and doors. We only take two types of them:
cupboard and desk, if the size of the cupboard is
120*210*70cm and the desk is 75*130*80 cm. The
selling price of each cupboard is around 325 dollars,
and the desk is around 170 dollars. Of course, to
make both kinds of products, there is a need for
different kinds of costs, such as the purchase of
wood, screws, knobs or pulls, drawer slides, hinges,
paint, etc. Thus, the cost of the cupboard is around
125 dollars and the cost of the desk is around 60
dollars, with losses, taxes, and rents (building rent,
water, and electricity rent) per week at around 300,
100, and 500 dollars, respectively. In this company,
there are two working groups (WG) to create the
production of the products: the carpenter group
(cutting and tying) and the polishing group (painting
and varnishing). Each group consisted of four
workers; each worker put in around 5 and 6 hours of
carpentry and polishing, except for Friday, which is
a holiday, and the fees of 10 and 5 dollars per hour,
respectively. Time required for the carpenter's
cupboard: 4 h and 2.5 h for the desk, and polish time
of 1.25 h and 0.75 h for the cupboard and desk. The
company's ability to produce both types is around 36
units per week. Make the above-mentioned problem
into a linear fractional programming formulation
under fuzzy coefficients and determine how many
cupboards and desks should be manufactured to
maximize the total profit.
Suppose that and are products for cupboards
and desks respectively, and arrange carpenters (C),
polishing (P), and job time (JT) work per week
together the fees per hour in tables (1) as:
Table 1. Total Fees and Hours.
WG
JT
L=W
G×JT
L×6
fees hour
fees
per
week
C
4
5
20
120
120×10
1200
P
4
6
24
144
144×5
720
1920
Profit function=income –cost function
Profit function=
Profit function=
To express the above problem in the form of an
FFLFP problem, we need to write it as follows:
Simultaneously, in the above problem there are
three constraints:
Carpenter time
Polishing time
Ability product of company
According to the values of the company assume that
Solution:
Considering our algorithm, we first solve the
problem as the usual LFP problem, as in the first
step. That is, we first express every coefficient as a
real number.
Step1:
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Problem 1
Subject to:
By Charnes and cooper’s method we have
Subject to:
Step2: By using the simplex method we obtain an
optimal solution
Step3: Convert the LFP problem into the FFLFP
problem
Subject to:
Step4: Convert the FFLFP problem into an LFP
problem using ().
Initially, we will use the first-ranking function
method, then
Therefore, the LFP problem is as follows:
Subject to:
Step5: Solve the LFP problem by repeating steps 1-
2, and then we get an optimal solution.
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The second time, we will use the second-ranking
function method),
,
Let, then
Similarly, for all fuzzy numbers of this example
applied that, we obtain the LFP problem as
follows:
Subject to:
By repeating step 1-2 in algorithm (5), we get
optimal solution
This time let . Then, the
, and the form of the problem is as
follows:
Subject to:
By repeating steps 1-2 in algorithm (5), we get an
optimal solution
To avoid lengthening the solution, we will indicate
the other values of where in the table
(2), where P1 is Problem 1 and FR is the first-
ranking function method.
Table 2. Shows the optimal solution to Problem 1 as
well as both ranking function methods.
N
o.
Total
profit
Exact
error
1
P 1
20
16
4867.59
2947.59
0
2
FR
30
2
5076.5
3156.5
-208.9
3
0
25
9
4935.72
3015.72
-68.13
4
0.125
24
11
4928.02
3008.02
-60.42
5
0.25
23
13
4899.55
2979.55
-31.96
6
0.375
22
15
4872.63
2952.63
-5.038
7
0.5
20
16
4867.59
2947.59
0
8
0.625
19
19
4843.65
2923.65
23.942
9
0.75
18
21
4821.43
2901.43
46.156
10
0.875
17
23
4780.93
2860.93
86.661
11
1
15
25
4763.05
2843.05
104.54
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7 Discussion
In table 2, it will be clear to us that the high and low
numbers of both types of products are different,
meaning that if the number of one of the products
made is larger, the other is smaller to reach the
optimal solution. At the same time, in the values of
w, we note that both products will make some of
them so that the company can make the most profits.
If we look at when the value of w=0.5 the solution is
the same as in Problem 1. The value of w in the
second-ranking function method is the most
profitable when it starts at zero and the least
profitable when it reaches one. And it's clear that
through the first ranking function method, the
problem is going to get the most profits and count
the best profit per week. To get the total profit, we
must subtract the (carpentry and polishing) fees
from the maximum value of the objective function.
And to see the effect and evaluation of both of our
ranking function methods, we needed to find the
exact rate of error compared to the main value of
Problem 1, and it made us realize that both ranking
function methods are very effective in analyzing and
reaching an optimal solution for each example of
TrFN of LFP problem. In the conclusion section, we
have explained how we can use our methods in
future work.
8 Conclusion
In this work, we have discussed the FFLFP problem
and shown a solution method that is very effective
and suitable for problems with fuzzy number
coefficients. Researchers have tried and solved it in
many ways. One of the ways is called the ranking
function method. In this study, we have proposed
two novel ranking function methods that are
effortless to calculate. At the same time, it is very
effective and useful because we have created a
living example in one of the companies in Erbil to
show the effectiveness of both types of ranking
methods. These ranking methods can be expanded
and used for other problems, such as linear
programming problems, quadratic programming
problems, and transportation problems in fuzzy
environments. In particular, the first ranking method
may be expanded to solve the FFLFP problem with
fuzzy numbers that are trapezoidal, pentagonal, and
hexagonal.
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_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.83
Rebaz Mustafa, Nejmaddin Sulaiman
E-ISSN: 2224-2880
717
Volume 21, 2022
