






























    
    

  

 KEYWORDS

Multiobjective
decision-making;
Intuitionistic fuzzy
decision set;
Goal
programming;
Interactive penalty
function.

ABSTRACT

Many applicable problems have multi-goals that optimize simultane-
ously, and decision-makers set imprecise aspiration levels for each goal.
Although such types of problems solved by fuzzy optimization are com-
mon in the literature, intuitionistic fuzzy optimization techniques are
more efficient to handle than fuzzy and classical optimization. This re-
search study focused on establishing a novel method by combining the
penalty function method with an interactive goal programming method-
ology for addressing multi-objective decision-making problems in an in-
tuitionistic fuzzy environment. One of the challenge that exists in the
literature of the optimization method under an imprecise decision en-
vironment is that it is not guaranteed to generate a Pareto-optimal so-
lution for the introduced problem. Therefore, in order to ensure the
Pareto-optimality of the obtained solution, the suggested method has
developed a new aggregation operator, an appropriate relaxation of the
constraint set, and a well-structured extended Yager membership func-
tion. In addition, unlike other methods in the literature, the suggested
method gives decision-makers the option to penalize the most unsatis-
fied objective function at a specific attained solution instead of starting
from scratch and working their way through the problem. To illustrate
the proposed method, we used a numerical example.

Many real-world decision-making prob-
lems such as agricultural cropland alloca-

tion problems by Moges et al. (2023a); Ba-
sumatary and Mitra (2022), transportation
problems by Tadesse et.al. (2023), water

∗

East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, Issue. 2, 41-57  |   

41

  

      
        

         

Demmelash Mollalign Moges1,∗,  Berhanu Guta Wordofa2,  Allen Rangia Mushi3

1Department  of  Mathematics,  Hawassa  University,  Hawassa,  Ethiopia
2Department  of  mathematics,  Addis  Ababa  University,  Addis  Ababa,  Ethiopia
3Department  of  mathematics,  University  of  Dares  Salaam,  Dares  Salaam,  Tanzania

     
     

Penalized Intuitionistic Fuzzy Goal Programming Method for Solving 
Multi-Objective Decision-Making Problems

1. INTRODUCTION

 
Research article

Corresponding author
Email address: danieldemelashalem@gmail.com +251913845944       https://dx.doi.org/10.4314/eajbcs.v5i2.4S



resource allocation problems by Sharma 
et.al. (2007), production planning decision-
making problems by Khan et.al. (2021), etc 
are defined as multi-objective programming 
problems (MOPPs). Often, the input data 
used in these problems does not have ex-
act numerical values because of various un-
controllable circumstances. To solve such 
types of MOPPs, the most appropriate and 
straightforward techniques are fuzzy opti-
mization methods, which were introduced 
by Bellman and Zadeh (1970). Many au-
thors (Tadesse et.al., 2023; Jameel and Radhi, 
2014; Kahraman et.al., 2016) have been more 
committed to apply fuzzy set tools for var-
ious application issues after Bellman and 
Zadeh (1970) provided a model to tackle 
fuzzy-multi-objective programming problems 
(FMOPPs).
The core idea behind the widely used ap-
proach for solving FMOPPs in the literature is 
to transform the original FMOPPs into a crisp 
single-objective optimization model using 
aggregation operators and ranking accuracy 
approaches, and then solve it using classical 
methods (Zimmermann, 2001; Kahra-man 
et.al., 2016; Bellman and Zadeh, 1970). Few 
papers have focused on developing new 
mathematical models for identifying fuzzy 
non-dominant solutions to FMOPP in the 
fuzzy optimization environment. These new 
models were developed by applying different 
aggregation operators like max-min operator 
by Bellman and Zadeh (1970), γ-operator by 
Zimmermann (2001), bounded min-sum op-
erator by Cheng et.al. (2013), and fuzzy-and 
operator by Singh and Yadav (2015) to con-
vert the multi-objective functions into single-
objection functions. Using the newly intro-
duced concepts, numerous approaches and 
models have been developed for various theo-
retical and scientific fields (Bogdana and Mi-
lan, 2009; Kassa and Tsegay, 2018; Tadesse

et.al., 2023). However, when uncertainty re-
sults from vagueness, inaccurate data, or in-
tentional judgments, the modeling capabili-
ties of fuzzy set theory are limited. Decision-
makers may also experience some hesitancy
as a result of imprecise information, un-
awareness of customers, seasonal change, etc.
These kinds of factors are critical to consider
while building realistic, suitable models and
solving decision-making problems (Singh and
Yadav, 2015; Razmi et.al, 2016; Moges and
Wordofa, 2024; Kumar, 2020; Fathy et.al.,
2023).

To overcome these limitations, numerous
researchers suggested various fuzzy exten-
sions that expanded the conventional fuzzy
set theory concepts. Atanassov (1986) de-
veloped a novel concept called an intuition-
istic fuzzy (IF) theory which successfully ad-
dresses the limitation of fuzzy theory. Since
the IF set can offer degrees of acceptance, hes-
itancy, and rejection, it has been found to be
more helpful for handling imprecision in op-
timization procedures than fuzzy and crisp-
based models, (Mollalign et.al., 2022; Sharma
et.al., 2023). Angelov (1995) proposed a new
model to determine the MN1 Pareto-optimal
solution based on intuitionistic fuzzy opti-
mization (IFO) which is a direct extension
of the fuzzy optimization technique put out
by Bellman and Zadeh (1970). Subsequently,
numerous scholars such as (Ghosh and Ku-
mar, 2014; Moges et al., 2023a; Rukmani and
Porchelvi, 2018a; Bharati and Singh, 2014;
Dey and Roy, 2015; Mollalign et.al., 2022;
Moges et.al., 2023b; Bharati et.al., 2014) have
been proposed different IFO methods to solve
the domain of MOPPs utilizing the benefit of
IFS tools.

Yager (2009) highlighted certain draw-
backs of Angelov (1995)’s method for de-
termining the optimal choice for decision-
makers. He suggested a new approach by con-

1MN represent Membership-Nonmembership

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42



verting the intuitionistic fuzzy decision envi-
ronment (IFDE) into a fuzzy decision envi-
ronment using a convex combination of non-
membership and membership functions. By
addressing the shortcomings of the Angelov
(1995) technique, Dubey et.al. (2012) ap-
plied the Yager (2009) strategy to solve the
MOPP. Garai et.al. (2015) exposes the short-
comings of the Dubey et.al. (2012) method
by demonstrating how certain constraints in
their model can impede the pursuit of the op-
timal solution or render the model imprac-
tical. They also suggest a new function by
broadening the scope of non-membership and
membership functions.

As with fuzzy optimization, a problem in
the IFO environment can be formulated as a
two-step process to be solved. The first is to
convert a multi-objective function into a crisp
single-objective optimization model using an
appropriate aggregation operator, and then
solve this model using suitable optimization
technique. The majority of the existing IFO
methods in the literature for resolving MOPP
are based on the max-min operator, (Moges
et al., 2023a; Dubey et.al., 2012; Garai et.al.,
2015). However, the max-min operator is not
guaranteed to generate the Pareto dominance
solution, and there is always no compensa-
tion for the resulting solution. The compen-
satory of the developed aggregation operator
has a critical role in the procedure of solving
MOPPs and it affects the optimality of the
resulting non-dominant solution. As a result,
the MN Pareto-solution might not be the re-
sult of a model that was developed using the
max-min operator approach.

The second limitation is that even if the
IFO method generates a MN Pareto-optimal
solution under an IFDE, there is not always
a guarantee of a Pareto-optimal solution to
the given MOPP due to the boundary prob-
lem. To overcome this difficulty, many re-
searchers have proposed different approaches
to fuzzy decision-making problems. But the
two-phase method which used by Lu et.al.

(2015); Dubois and Fortemps (1999); Wu and
Guu (2001); Dubey et.al. (2012); Tsegaye
et.al. (2021); Jiménez and Bilbao (2009) is the
most commonly applied approach for finding
a Pareto-optimal solution to MOPPs in fuzzy
decision environment. A two-phase method
means that in phase one, use the max-min
operator technique and then use the mean-
operator technique in phase two to improve
the previous solution obtained by the max-
min operator approach. However, Dubois
and Fortemps (1999) indicated that this kind
of strategy is not entirely consistent because
we need to switch from the max-min opera-
tor to the mean-operator at different stages,
and they suggested a multiphasons for objec-
tive functions provides upper and lower tol-
erances to avoid decision deadlock. Addi-
tionally, Razmi et.al (2016) point out that
in situations where a certain level of satis-
faction is fully achieved, there might not be
a guarantee that a fuzzy efficient solution
is Pareto-optimal. In order to generate the
Pareto-optimal solution, Jiménez and Bilbao
(2009); Razmi et.al (2016) extended the tech-
nique by Wu and Guu (2001) and developed
a generic strategy based on the goal program-
ming method. On the other-hand, Mollalign
et.al. (2022) demonstrate that there is no as-
surance that the imprecise environment ap-
proach put forward by Razmi et.al (2016) will
be the only method used to obtain the Pareto-
optimal solution of MOPP when using intu-
itionistic fuzzy hierarchical optimization.

Furthermore, the IFO approach offers up-
per and lower tolerances to prevent decision
stalemate when defining membership/ satis-
faction and nonmembership/ dissatisfaction
for objective functions. The third limitation
in the literature of IFO is that to determine
upper and lower tolerance, researchers use
the “payoff matrix” approach without con-
sideration of underestimation or overestima-
tion values of the nadir point. Therefore,
decision-makers may perceive the solution de-
rived from this approach incorrectly. But the

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43



interactive approach is helpful to a decision-
maker since it offers mechanisms for learn-
ing about a problem in the IFO. There are
a few researchers that concern an interac-
tive method for solving a different domain of
MOPPs in an intuitionistic fuzzy decision en-
vironment, (Hanine et.al., 2021; Garai et.al.,
2016). However, in the existing interactive
method, decision-makers do not have the free-
dom to punish/ penalize the more unsatis-
fied objective function at each iteration when
the current solution does not satisfy the DM,
rather than solving the problem from scratch.

By taking into account the aforemen-
tioned limitations, the key goal of this
research study is to develop a general
and powerful novel method for solving the
multi-objective linear decision-making prob-
lem (MOLDMP) in an IFDE using the com-
bination of penalty function method, the in-
teractive method, and the goal programming.
Additionally, the following points are specific
and basic contributions of current research-
study works:

i. We formulated an IF membership and
nonmembership function by finding the
correct ideal and nadir point to set
boundary for intuitionistic fuzzy goals
(IFGs).

ii. We proposed an extended Yager-
membership function that eliminates
the boundary value problem.

ii. A new IF aggregation operator is de-
veloped to convert MOLDMP into a
single-objective decision-making prob-
lem.

iii. Using a new IF aggregation opera-
tor, we proposed an IFG programming
model to find a Pareto-optimal solution
for MOLDMP.

iv. Decision-makers provide an interactive
penalty function method to punish un-

satisfactory objective functions at the
current solution.

The rest of the paper is presented as fol-
lows: the basic concepts and terms related
to the paper are presented in Section 2. Sec-
tion 3 illustrates the mathematical formula-
tion of IF-MOLDMPs and its IFG model. In
Section 4, we demonstrated the newly pro-
posed method developed based on goal pro-
gramming techniques and interactive penalty
function method. The general framework or
algorithm of the proposed method presented
in Section 5 and the numerical examples used
to summarize and demonstrate the applica-
bility of the introduced method are discussed
in Section 6. The results and discussion of
the study are given in Section 7. Finally, the
conclusion of the present work and its future
scope are given in Section 8.

2. PRELIMINARIES

2.1. Intuitionistic Fuzzy Set

Definition 2.1. (Atanassov, 1986; Fathy
et.al., 2023) An intuitionistic fuzzy set
(IFS) ÃI in a non-empty universal X
is a set of ordered-triplets ÃI = {<
x, µÃI (x), νÃI (x) >: x ∈ X} where, µÃI :

X → [0, 1] represent the membership func-
tion or degree of belongingness and νÃI :
X → [0, 1] represent non-membership func-
tion or degree of non-belongingness of the el-
ement x ∈ X being in ÃI , so that ∀x ∈ X,
0 ≤ µÃI (x) + νÃI (x) ≤ 1,. For any IFS ÃI on
X, πÃI (x) = 1−µÃI (x)−νÃI (x) is the degree
of indeterminacy of x ∈ ÃI or x /∈ ÃI .

2.1.1. Operations over Intuitionistic Fuzzy
Sets

Assume that ÃI = {< x, µÃI (x), νÃI (x) >
|x ∈ X} and B̃I = {< x, µB̃I (x), νB̃I (x) >
|x ∈ X} are any IFSs on the universal set
X, (Husain et.al., 2012).

1. subset ÃI ⊆ B̃I ⇐⇒ µÃI (x) ≤ µB̃I (x)
and νÃI (x) ≥ νB̃I (x), ∀x ∈ X.

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44



2. Equal Set: ÃI = B̃I ⇐⇒ µÃI (x) =
µB̃I (x) and νÃI (x) = νB̃I (x), ∀x ∈ X.

3. Complementation: (ÃI)c = {<
x, νÃI (x), µÃI (x) > |x ∈ X}.
To represent the minimum and maxi-
mum operator (that means: min and
max operator), we use the symbols “∧”
and “∨” respectively.

4. Intersection: ÃI ∩ B̃I = {< x, µÃI (x) ∧
µB̃I (x), νÃI (x) ∨ νB̃I (x) > |x ∈ X}.

5. Union: ÃI ∪ B̃I = {< x, µÃI (x) ∨
µB̃I (x), νÃI (x) ∧ νB̃I (x) > |x ∈ X}.

2.2. Multi-objective programming
problem

Assume that a vector-valued objec-
tive function f : ℜn → ℜk (f(x) =
(f1(x), f2(x), f3(x) . . . , fk(x)), and constraint
functions g : ℜn → ℜm (g(x) =
(g1(x), g2(x), g3(x), . . . , gm(x)) are continu-
ously differentiable for all decision vector x =
(x1, x2, x3 . . . , xn) ∈ S is formulated in the
following way: (Tsegaye et.al., 2021; Razmi
et.al, 2016)

min f(x) = (f1(x), f2(x), f3(x) . . . , fk(x))
T

Subject to: x ∈ S =

{
x ∈ ℜn gj(x) (≤,≥,=) 0, j = 1, 2, . . . ,m

xi ≥ 0, i = 1, 2, . . . , n.

}
(1)

Definition 2.2. (Li and Hu, 2009) Let a vec-
tor x∗ ∈ S be a feasible solution to MOPP
(1). Then

� x∗ is weakly Pareto-optimal solu-
tion to MOPP (1) if there doesn’t exist
a new vector x ∈ S such that ft(x) <
ft(x

∗) for all t = 1, 2, . . . , k.

� x∗ is Pareto-optimal (efficient) so-
lution to MOPP (1) if there doesn’t
exist a new vector x ∈ S such that
ft(x) ≤ ft(x

∗) for all t = 1, 2, . . . , k,
and ft(x) < ft(x

∗) for some t.

Any Pareto-optimal solution is weakly
Pareto-optimal but the converse is hold for
convex optimization problem. Assume that

the feasible set S ̸= Ø (or Z = f(S) ̸= Ø) is
compact and ft(x),∀t, is continuous to unsure
the Pareto-optimal is exist for MOPP (1).

3. MATHEMATICAL FORMULA-
TION OF PROBLEM

3.1. Intuitionistic Fuzzy Multi-
objective Linear Decision-Making
Problem

Consider mathematical formulation of
Intuitionistic Fuzzy Multi-objective Linear
Decision-Making Problem (IF-MOLDMP)
can be represented as: (Rukmani and
Porchelvi, 2018a; Basumatary and Mitra,
2022; Garai et.al., 2015)

m̃ax (m̃in) f̃ I
t (x) = Cx+ d, t = 1, 2, . . . , k

Subject to: x ∈ S̃I = {x ∈ ℜn : g̃Ii (x) (≤̃, ≥̃,≈) bi,x ≥ 0},
(2)

where C ∈ ℜk×n, dT ∈ ℜk are determin-
istic parameters and the right-hand quantity
is bi ∈ ℜ, i = 1, 2, . . . ,m. The IF-version of
classical inequality≤,≥ and equality = in the
model (1) are given as ≤̃, ≥̃ and ≈, respec-
tively. The function f̃ I

t (x) and g̃Ii (x) are IF
linear objective and constraints, respectively
due to they have IF aspiration levels set by
decision-makers (DMs). S̃I is IF feasible con-

vex region.

3.2. Intuitionistic Fuzzy Goal Model of
IF-MOLDMP

In many real-life applications, the
decision-maker is allowed to specify an
imprecise aspiration-level for each of the
constraint and objectives function in IF-
MOLDMP (2). A goal with an imprecise

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45



aspiration-level for an IF objective and con-
straints can be treated as an intuitionistic
fuzzy goal (IFG) in the IFDE. Let D̃I =
{f̃ I

1 (x), f̃
I
2 (x), . . . , f̃

I
k

S} be a set of IFGs, having maximization
(M1) with aspiration-level f̄t for t ∈ M1,
ḡi for i ∈ N1 and minimization (M2) with

aspiration-level f
t
for t ∈ M2, gi for i ∈ N2

for type of IFGs. Hence, to determine the
crisp/ deterministic optimal solution, we need

(x), g̃1
I (x), g̃2

I (x), . . . , g̃Im(x)|x ∈            to formulate the IFG model of IF-MOLDMP
(2) as: (Moges et al., 2023a; Razmi et.al,
2016)

Determine: x ∈ S̃I

Subject to:

{
ft(x) ≥̃ f̄t, gi(x) ≥̃ ḡi for t ∈ M1, i ∈ N1

ft(x) ≤̃ f
t
, gi(x) ≤̃ g

i
for t ∈ M2, i ∈ N2

(3)

In order to find a  P areto-optimal so-
lution for IF-MOLDMP (2) in the IFDE, 
we need to determine a solution that si-
multaneously maximizes the level of satis-
faction/ membership µt(ft(x)), µi(gi(x)) and 
minimizes the level of dissatisfaction/ non-
membership νt(ft(x)), νi(gi(x)) of the IFGs 
under the given feasible region, x ∈ S, ac-

cording to Angelov (1995) stated. Based on
this idea, the Pareto-optimal and MN Pareto-
optimal solution of IF-MOLDMP (2) are de-
fined as follows:

Definition 3.1. (Razmi et.al, 2016; Tsegaye
et.al., 2021) A solution vector x∗ ∈ S̃I is said
to be a Pareto-optimal to IF-MOLDMP (2)
if there is no new-vector x ∈ S̃I such that

f̃ I
t (x)≤̃f̃ I

t (x
∗) ∧ µi(gi(x)) ≥ µi(gi(x

∗)) ∧ νi(gi(x)) ≤ νi(gi(x)
∗),∀t = 1, 2, . . . , k,∀i = 1, 2, . . . ,m,

f̃ I
t (x)<̃f̃ I

t (x
∗) ∨ µi(gi(x)) > µi(gi(x

∗)) ∨ νi(gi(x)) < νi(gi(x)
∗), for some t ∈ {1, 2, . . . , k}, i ∈ {1, 2, . . . ,m}.

Definition 3.2. (Jafarian et.al., 2018; Razmi
et.al, 2016) A solution vector x∗ ∈ S̃I

is said to be a MN Pareto-optimal solu-
tion to IF-MOLDMP (2) if there doesn’t
exists another vector x ∈ S̃I such that
µt(ft(x)) ≥ µt(ft(x

∗)) ∧ νt(ft(x)) ≤
νt(ft(x)

∗) ∧ µi(gi(x)) ≥ µi(gi(x
∗)) ∧

νi(gi(x)) ≤ νi(gi(x)
∗),∀t = 1, 2, . . . , k,∀i =

1, 2, . . . ,m, and strictly inequality holds for
some t or i.

4. PROPOSED SOLUTION METHOD

4.1. Extended Yager-membership
function in the IFDE

In the IFDE, the nadir and ideal points
are useful for estimating the range of de-
grees of satisfying (membership) and rejec-
tion (non-membership) for the objective func-
tion in the IF-MOLDMP (2). To determine

nadir and ideal points, we need to find the
optimal solution for each objective function
ft(x) subject to the given set of constraints.
That means solving the following problem in-
dependently:

min ft(x) Subject to x ∈ S

for t = 1, 2, 3, . . . , k
(4)

The solution obtained from model (4) we call
as xB

t is best solution, ZB
t = ft(x

B
t ) is ob-

jective value or aspiration-level for each t.
Zid = (ZB

1 , Z
B
2 , . . . , Z

B
k ) is an ideal point and

Znad = (Z̄1, Z̄2, . . . , Z̄k) is nadir point of IF-
MOLDMP (2) where, Z̄t = max

x∈P ∗
ft(x), P

∗ is

Pareto-optimal set. Since P ∗ ⊂ S, we have
max
x∈S

ft(x) ≥ Z̄t,∀t = 1, 2, 3, . . . k.

Due to the difficulties in finding the nadir
point, some scholars suggested a heuristic-
approach called “Payoff matrix”, but the re-

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46



sult obtained by this approach may be an un-
derestimation or overestimation of the nadir
point, Isermann and Steuer (1997). The best
solution x̄B

t is used to construct the payoff
matrix as follows:

f1(x) f2(x) . . . fk(x)
xB
1 f1(x

B
1 ) f2(x

B
1 ) . . . fk(x

B
1 )

xB
2 f1(x

B
2 ) f2(x

B
2 ) . . . fk(x

B
2 )

...
...

...
. . .

...
xB
k f1(x

B
k ) f2(x

B
k ) . . . fk(x

B
k )

 (5)

If the best solutions xB
t are unique for all

t = 1, 2, 3, . . . , k, then the payoff matrix ap-
proach will never overestimate the nadir point
Znad, (Isermann and Steuer, 1997). How-
ever, if xB

t not unique for at least one t,
max{ft(xB

1 ), ft(x
B
2 ), . . . , ft(x

B
k )} may overes-

timate the nadir point. When a component
of the nadir point for an optimization prob-
lem is underestimated or overestimated in an
IFDE, it may result in unnecessary informa-
tion about the Pareto-optimal solution, and
decision-makers use a wide-range.

If the model (4) produces more than one
best solution for any s ∈ {1, 2, . . . , k}, then
underestimated or overestimated solutions of
nadir point exist. To estimate the correct
nadir point, we use two-steps in the proposed
method. In step one, we solve the follow-
ing optimization problem for each s, using
σ = 1

k
> 0.

min
fs(x)

ZB
s

+ σ

k∑
t̸=s

ft(xt)

ZB
t

Subject to: x ∈ S

(6)

Let xBB
s be a solution obtained from the

model (6), and construct the Payoff-matrix
(5) using this solution instead of xB

s . Using
this two-step approach, we find upper and
lower-bounds for the membership and non-
membership functions of the objective func-
tion, which are used as degrees of rejection
and acceptance of each IFGs.

In the case of minimization problem,
Uµ
t = max{ft(xB

1 ), ft(x
B
2 ), . . . , ft(x

B
k )} and

Lµ
t = min{ft(xB

1 ), ft(x
B
2 ), . . . , ft(x

B
k )} con-

sidered as upper and lower-bound of mem-
bership functions, respectively. Similarly,
U ν
t = Uµ

t and Lν
t = Lµ

t + ϵt(U
µ
t − Lµ

t ) where,
0 < ϵt < 1 for each t are upper and lower-
bound of non-membership functions, respec-
tively (Bharati and Singh, 2014; Moges et.al.,
2023b).

Lemma 4.1. The solution obtained from the
model (6) never generates an overestimation
of Znad, (Ehrgott, 2000; Isermann and Steuer,
1997).

For minimization problem, membership
(µt(ft(x))) and non-membership (νt(ft(x)))
functions can be defined according to Eqs. 7
and Eqs. 8 respectively.

µt(ft(x)) =


1 if ft(x) ≤ Lµ

t
Uµ
t −ft(x)

Uµ
t −Lµ

t
if Lµ

t < ft(x) ≤ Uµ
t

0 if ft(x) > Uµ
t

(7)

νt(ft(x)) =


0 if ft(x) < Lν

t
ft(x)−Lν

t

Uν
t −Lν

t
if Lν

t ≤ ft(x) ≤ U ν
t

1 if ft(x) > Uν
t

(8)
Now, to overcome the limitation of the An-
gelov (1995) model, we first resolve the inde-
terminacy factors independently for each IFG
using the Yager (2009) approach. Therefore,
using the membership function µt(ft(x)) de-
fined in Eqs.7 and the non-membership func-
tion νt(ft(x)) defined in Eqs.8, the new ex-
tended Yager-membership function is defined
as follows: for any λ ∈ [0, 1].

Iλt (ft(x)) = (1− λ)µt(ft(x)) + λ(1− νt(ft(x)))

for each t = 1, 2, . . . k.
(9)

Without knowing any other details regard-
ing the DM’s attitude during this study,
we arrived at λ = 1

2
in our discussion

that follows. The piece-wise linear Yager-
membership function for the minimization

East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 41-57

47



problem shown in Figure 1(a): and de-
fined as:

Iλt (ft(x)) =


1 if ft(x) ≤ Lµ

t

1− 1
2
ϵt(

ft(x)−L
µ
t

Lν
t −L

µ
t

) if Lµ
t ≤ ft(x) < Lν

t

1
2
(2− ϵt)

U
µ
t −ft(x)

U
µ
t −Lν

t
if Lν

t ≤ ft(x) < Uµ
t

0 if ft(x) ≥ Uµ
t

(10)

Now, we need to expand the range of function
Iλt (ft(x)) = 1, for ft(x) ≤ Lµ

t to Iλt (ft(x)) >
1, for ft(x) < Lµ

t to overcome the limitation
of Dubey et.al. (2012) model.

Suppose Lmin
t = min{Lµ

1 , L
µ
2 , . . . , L

µ
k} ±

Lµ
t and Umax

t = max{Uµ
1 , U

µ
2 , . . . , U

µ
k } ± Uµ

t

such that Lmin
t < Lµ

t ≤ ft(x) ≤ Uµ
t for

each t = 1, 2, . . . , k. To extend the Yager-
membership function, choose two positive
numbers, b1, b2 > 0, and relax the range from
[0, 1] to [−b2, 1+ b1]. Therefore, the extended
Yager-membership function for the minimiza-
tion problem developed as shown in Figure
1(b): and defined in the following way:

ηt(ft(x)) =



b1(L
µ
t −ft(x))

Lµ
t −Lmin

t
+ 1 if Lmin

t < ft(x) < Lµ
t

1− 1
2
ϵt(

ft(x)−Lµ
t

Lν
t −Lµ

t
) if Lµ

t ≤ ft(x) < Lν
t

1
2
(2− ϵt)

Uµ
t −ft(x)

Uµ
t −Lν

t
if Lν

t ≤ ft(x) < Uµ
t

b2(ft(x)−Uµ
t )

Uµ
t −Umax

t
if Uµ

t ≤ ft(x) ≤ Umax
t

(11)

Figure 1: Yager-membership function Iλt (ft(x)), (Aggarwal et.al., 2019) and Extended Yager-membership func-
tion ηt(ft(x)), (Garai et.al., 2016) for the minimization problem

The piecewise linear extended Yager-
membership function Eqs. (11) has the fol-
lowing properties:

� Over the interval containing all possible
values of the objective function, it is a
strictly monotonic function.

� It permits alternate point orderings
with Yager-membership function values
outside of [0, 1].

� It is equivalent to the Yager-
membership function Iλt (ft(x)) Eqs.
(10) on [Lµ

t , U
µ
t ] for each t.

� If ft(x) < Lµ
t , then ηt(ft(x)) ≥ 1, and

if ft(x) > Uµ
t , then ηt(ft(x)) ≤ 0.

� It shows the level of satisfaction of the
decision-maker for every objective func-
tion.

Additionally, we modified these piecewise

48

East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 41-57



linear extended Yager-membership functions
ηt(ft(x)) as follows: based on the method sug-
gested by Wu and Guu (2001) to solve MOPP

(1) under an IFDE without introducing extra
binary variables.

ηt(ft(x)) = ηt(a
1
t ) + s1t (ft(x)− a1t ) +

m−1∑
l=2

(slt − sl−1
t )

2
(|ft(x)− alt|+ ft(x)− alt) (12)

where, alt, l = 1, 2, 3, . . . ,m are the break-
ing/ jumping points of ηt(ft(x)), t = 1, . . . , k,
the slope of line segment between alt and a

l+1
t

is given by slt =
ηt(a

l+1
t )−ηt(alt)

al+1
t −alt

for l =

1, . . . ,m− 1; t = 1, 2, . . . , k.

4.2. Develop a new intuitionistic fuzzy
(IF) aggregation operator

A number of logical operators are avail-
able in the IFDE, enabling the combination
of numerous objectives into a single objective.
We employed the convex combination of the
arithmetic mean-operator and the max-min
operator on the extended Yager-membership
functions ηt(ft(x)) Eqs. (11) in order to cre-
ate a novel operator in the IFDE. Therefore,
the new IF aggregation operator is formulated
as:

ηD̃I (x) = δ mink
t=1ηt(ft(x))+(1−δ)

1

k

k∑
t=1

ηt(ft(x)),

(13)

where 0 < δ < 1 represent the degree of pref-
erence of DMs. Using this novel IF aggrega-
tion operator and decision-makers’ preference 
value δ ∈ [0, 1], the IF-MOLDMP (2) is trans-
formed into a crisp single-objective optimiza-
tion problem to determine the Pareto-optimal 
solution based on Eqs.(14).

max
x∈S

ηD̃I (x) (14)

Therefore, using a non-negative variable
qlt, the extended Yager-membership function
defined in Eqs.(12), and the IF aggregation
operator defined in (13), the equivalent deter-
ministic single-objective optimization prob-
lem of model (14) is formulated as follows:

Model I:

max δα0 + (1− δ)
k∑

t=1

αt

Subject to:


α0 + αt ≤ ηt(ft(x)) for t = 1, 2, . . . , k.
ηt(ft(x)) = ηt(a

1
t ) + s1t (ft(x)− a1t ) + · · ·+ (sm−1

t − sm−2
t )(ft(x)− am−1

t + qm−2
t )

ft(x) + ql−2
t ≥ al−1

t for l = 3, 4, . . . ,m
α0, αt, q

l−2
t ≥ 0 for t = 1, 2, . . . , k; l = 3, 4, . . . ,m.

x ∈ S


(15)

4.3. The intuitionistic fuzzy goal pro-
gramming method

The higher-value of ηt(ft(x)) is regarded
as the best acceptable value for DMs in
the IFDE. As a result, DMs must minimize

the under-achievement (negative-deviational)
variable d−t by assigning weight w−

t to each
objective function t in the goal programming
approach ηt(ft(x)) + d−t ≥ 1. Therefore, the
intuitionistic fuzzy goal programming (IFGP)

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49



model of IF-MOLDMP (2) is formulated as
follows:

Model II:

min
k∑

t=1

w−
t d

−
t

Subject to:


ηt(ft(x)) + d−t ≥ 1 for t = 1, 2, 3, . . . , k
x ∈ S

d−t ≥ 0,
∑k

t=1w
−
t = 1, w−

t > 0


(16)

4.4. Interactive Penalty Function
Method (IPFM)

In an intuitionistic fuzzy decision environ-
ment (IFDE), the interactive approach is very
effective at solving IF-MOLDMP (2). In this
process, the algorithm generates an initial so-
lution before consulting the DM and obtain-
ing a new solution(s) if the DM is dissatisfied
with the current one. Let x∗ be the optimal
solution of model I (15) or model II (16). As-
sume that the existing solution x∗ does not
satisfy the DMs.

To discuss how the penalty function

method is applied in the interactive method
once DMs update the problem for some ob-
jective functions, we considered the following
Figure 2 that shows the level of satisfaction
and dissatisfaction by DMs. Furthermore,
the Figure 2 demonstrates how to determine
a solution from a large space by relaxing the
constraint set. The green region in the Figure
2 represents DMs who are completely satisfied
with the current solution, while the red region
represents DMs who are dissatisfied, and the
row depicts how the algorithm generates the
best solution from a large space.

Figure 2: How penalty function method applied on minimization type problem

Definition 4.1. A vector x ∈ S
is DM-feasible solution if ∀t ∈
1, 2, . . . , k, ηλ

D̃I (ft(x)) − ηλ
D̃I (ft(x

∗)) ≥ γt, for
nonnegative variable γt. Where, x∗ is optimal
solution to either model I (15) or model II
(16).

Let the set of DM-feasible solution be de-
noted by SDM and defined as SDM = {x ∈
S|ηλ

D̃I (ft(x))− ηλ
D̃I (ft(x

∗)) ≥ γt,∀t}.

To find a preferable Pareto-optimal solu-
tion based on x∗ to IF-MOLDMP (2), we

East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 41-57

50



solve the following optimization problem:

max
x∈SDM ,γt≥0

j∑
t=1

γt (17)

Now choose x ∈ S such that ηλt (ft(x)) −
ηλt (ft(x

∗)) < γ2
t (i.e.,x /∈ SDM). The non-

linear penalty function is defined as P (x) =∑j
t=1[max{0, ηλ

D̃I (ft(x
∗))−ηλ

D̃I (ft(x))+γ2
t }]2

Meng et.al. (2011); Jameel and Radhi (2014)
and satisfies:

P (x) =

{
0 if x ∈ SDM

> 0 if x /∈ SDM (18)

Now, using the penalty function, we can
transform it into unconstrained optimization,
where the objective function is defined as:
F (x, ci) =

∑j
t=1 γt+ci(

∑j
t=1[max{0, ηλt (ft(x∗))−

ηλt (ft(x)) + γ2
t }]2)

Based on the penalty parameter ci > 0, we
formulate the penalty optimization problem
as follows:

(Pλ,ci) maxF (x, ci)

Subject to x ∈ Rn,
(19)

The main benefit of this model is that, in-
stead of starting from scratch, it will use a
wide domain to improve the existing solution
and attempt to discover a solution that sat-
isfies DMs by ignoring the non-update mem-
bership and non-membership functions.
IPFM Algorithm:
Initial Step:

� Using current solution x∗ and γti =

ηλ
D̃I (

Uµ
t −Lµ

t

2
), choose initial solution x1 ∈

S such that ηλt (ft(x1)) − ηλt (ft(x
∗)) <

γ2
ti
. That means: x1 /∈ SDM .

� Choose λ = 1
2
, c1 > 0, β > 1 and set i=1

Main Step:

Step I. Using xi and γti solve the problem
maxF (x, ci) Subject tox ∈ Rn and
let call xi+1 be an optimal solution.

Step II. If xi+1 be DM feasible solution, then
stop x̂ = xi+1 and the corresponding
decision vector γ̂ are optimal solutions
to model (17). Otherwise, put ci+1 =
βci, i = i+ 1 and go to step I.

Theorem 4.1 (Optimality test). Let
x̂ and γ̂ are optimal solutions to model (17).
Then

i. If γt = 0,∀ t = 1, . . . k, then the Pareto
optimal solution to IF-MOLDMP (2) in
an IFDE is x∗.

ii. In an IFDE, x∗ is not Pareto’s optimal
solution to IF-MOLDMP (2) if at least
one γt > 0. Instead of x∗, Pareto’s op-
timal solution to IF-MOLDMP (2) is x̂.

(Analogous theorem proofs can be found in
literature for instance see Garai et.al. (2015,
2016))

5. ALGORITHM FOR PENALIZED
IFGP METHOD

Based on the idea discussed above, we
proposed a general framework or algorithm
for finding Pareto-optimal solutions for IF-
MOLDMP (2) in an IFDE. The proposed al-
gorithm’s steps are as follows:

Step 1: Solve each objective function indepen-
dently under the constraint set. That
means: solve model (4).

Step 2: If the solution xB
t of model (4) is unique

for each t, then go to step 3. Otherwise,
solve model (6) and use its solution xBB

t

instead of xB
t , then go to step 3.

Step 3: Construct the payoff matrix (see Eqs.
5) and find the upper and lower-
tolerances of membership and non-
membership functions.

Step 4: Formulate the extended Yager-
membership function in an intuition-
istic fuzzy decision environment. See
Eqs. (11).

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51



Step 5: Solve either model I (15) or model II
(16).
If the decision-maker is satisfied with
the current solution x∗, then stop, and
the current solution x∗ is the Pareto
optimal solution to IF-MOLDMP (2).
Otherwise, go to step 6.

Step 6: Ask the decision-maker to change the
membership and nonmembership func-
tions for j ≤ k objective functions, then
go to step 7.

Step 7: Solve model 17 with the above-
mentioned algorithm for interactive
penalty function method to get solu-
tions x̂ and γt. Then there are the fol-

lowing scenarios:

Case-I If γt = 0,∀ t = 1, . . . k, decision-
makers must adjust either δ for
model I and weight of objective
function for model II or λ, then go
to step 5.

Case-II In an IFDE, if at least one γt > 0,
then Pareto’s optimal solution to
IF-MOLDMP (2) is x̂.

6. NUMERICAL EXAMPLE

Consider the following intuitionistic fuzzy
multi-objective linear decision-making prob-
lem (IF-MOLDMP) that is given in an intu-
itionistic fuzzy decision environment:

m̃ax f1(x) = −x1 + 2x2

m̃ax f2(x) = 2x1 + x2

Subject to: x ∈ S =


−x1 + 3x2 ≤ 21, 4x1 + 3x2 ≤ 45,

x ∈ R2 x1 + 3x2 ≤ 27, 3x1 + x2 ≤ 30,
x1, x2 ≥ 0.


(20)

Following the proposed solution method
discussed so far, we solve the given IF-
MOLDMP (20) step by steps:

� Individual Solution: xB
1 = (0, 7),xB

2 =
(9, 3), and Ideal point: (ZB

1 , Z
B
2 ) =

(14, 21)

� Using Payoff matrix the upper and
lower bounds are: Uµ

1 = 14, Lµ
1 =

−3, Lmin
1 = −6, Umax

1 = 27 and Uµ
2 =

21, Lµ
2 = 7, Lmin

2 = −10, Umax
2 = 32 us-

ing a tolerances ϵ1 = 0.4, ϵ2 = 0.3.

� The brake points are:
a11 = −6, a21 = −3, a31 = 7, a41 =
14, a51 = 27 and its corresponding values
are η1(a

1
1) = −b2, η1(a

2
1) = 0, η1(a

3
1) =

0.8, η1(a
4
1) = 1, η1(a

5
1) = 1 + b1 for ob-

jective function f1(x).
a12 = −10, a22 = 7, a32 = 17, a42 =
21, a51 = 32 and its corresponding values
are η2(a

1
2) = −b2, η2(a

2
2) = 0, η2(a

3
2) =

0.85, η2(a
4
2) = 1, η2(a

5
2) = 1 + b1 for ob-

jective function f2(x).

� The extended Yager-membership func-
tions are:

η1(f1(x)) = −0.0286x1 + 0.057x2 − 0.59q11
− 0.0514q21 + 0.61 and

η2(f2(x)) = 0.074x1 + 0.037x2 − 0.033q12
− 0.048q22 + 0.227

� Based on the proposed model (15), we
formulated IF-MOLDMP (20) as single-
objective optimization model:

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52



Model I: Assign the value of δ ∈ [0, 1]

max δα0 + (1− δ)α1 + (1− δ)α2

Subject to



α0 + α1 + 0.0286x1 − 0.057x2 + 0.59q11 + 0.0514q21 ≤ 0.61
α0 + α2 − 0.074x1 − 0.037x2 + 0.033q12 + 0.048q22 ≤ 0.227
x1 − 2x2 − q11 ≤ 3, x1 − 2x2 − q21 ≤ −7
−2x1 − x2 − q12 ≤ −7, −2x1 − x2 − q22 ≤ −17
−x1 + 3x2 ≤ 21, 4x1 + 3x2 ≤ 45, x1 + 3x2 ≤ 27, 3x1 + x2 ≤ 30
x1, x2, α0, α1, α2, q

1
1, q

2
1, q

1
2, q

2
2 ≥ 0.


(21)

� In a similar way, using the proposed model (16), we converted the given IF-MOLDMP
(20) into the following single-objective optimization model:

Model II: Assign the relative important of each objective function w−
t

minw−
1 d

−
1 + w−

2 d
−
2

Subject to:



0.0286x1 − 0.057x2 + 0.59q11 + 0.0514q21 − d−1 ≤ −0.39
−0.074x1 − 0.037x2 + 0.033q12 + 0.048q22 − d−2 ≤ −0.773
x1 − 2x2 − q11 ≤ 3, x1 − 2x2 − q21 ≤ −7
−2x1 − x2 − q12 ≤ −7, −2x1 − x2 − q22 ≤ −17
−x1 + 3x2 ≤ 21, 4x1 + 3x2 ≤ 45, x1 + 3x2 ≤ 27, 3x1 + x2 ≤ 30
x1, x2, d

−
1 , d

−
2 , q

1
1, q

2
1, q

1
2, q

2
2 ≥ 0.


(22)

Now, using MATLAB-R2023 software
to solve model I (21) and model II (22),
we obtained the Pareto-optimal solution

x1 = 6,x2 = 7, fval = 1.1311, f1(x
∗) = 8,

and f2(x
∗) = 19 for the given IF-MOLDMP

(20) and shown in Table 1.

Table 1: Results for model I (21) and model II (22) in different cases

δ, w−
t Optimal Solution Optimal Value

Model I
δ = 0.36 α0 = 0,α1 = 0.83739,α2 = 0.929,

x1 = 6,x2 = 7, fval = 1.1311
f1(x

∗) = 8,
f2(x

∗) = 19
δ = 0.5 α0 = 0,α1 = 0.8374,α2 = 0.93, x1 =

5.991, x2 = 7, fval = 0.8837
f1(x

∗) = 8.009,
f2(x

∗) = 18.982
δ = 0.8 α0 = 0.8777,α1 = 0,α2 = 0, x1 =

5.152, x2 = 7.282, fval = 0.7022
f1(x

∗) = 9.412,
f2(x

∗) = 17.586

Model II
w−

1 = 0.6, w−
2 = 0.4 d−1 = 0.10548,d−2 = 0.144, x1 =

4.800,x2 = 7.399,fval = 0.12089
f1(x

∗) = 9.998,
f2(x

∗) = 16.882
w−

1 = 0.5, w−
2 = 0.5 d−1 = 0.1626,d−2 = 0.0700, x1 =

6.00,x2 = 6.999,fval = 0.1163
f1(x

∗) = 7.998,
f2(x

∗) = 18.999
w−

1 = 0.3, w−
2 = 0.7 d−1 = 0.1626,d−2 = 0.0700,

x1 = 6,x2 = 7.00, fval = 0.0978
f1(x

∗) = 8,
f2(x

∗) = 19

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53



7. RESULTS AND DISCUSSION

For different values of δ ∈ (0.35, 0.85) and
w−

1 , w
−
2 ≥ 0 with w−

1 + w−
2 = 1, the compar-

ative work of the resulted efficient solution
is given in Table 1. Let the optimal or ob-
jective value of model I (21) and model II
(22) be presented by fval. As shown in Ta-
ble 1, when the value of w−

2 (relative-weight
of f2(x)) increases and the value of δ (de-

gree of compensation), w−
1 (relative-weight of

f1(x)) decreases, the preferable optimal value
fval is obtained. Thus, the optimal solution
of both model I (21) and model II (22) is
identical, i.e., x1 = 6,x2 = 7, fval = 1.1311,
which is the candidate Pareto-optimal solu-
tion of IF-MOLDMP (20). Now to test the
Pareto-optimality, we need to formulate the
following single-objective problem (23) based
on Eqs. (17) and solve it:

max γ1 + γ2

Subject to:



−0.0286x1 + 0.057x2 − 0.59q11 − 0.0514q21 − γ1 ≥ 0.2274
0.074x1 + 0.037x2 − 0.033q12 − 0.048q22 − γ2 ≥ 0.703
x1 − 2x2 − q11 ≤ 3, x1 − 2x2 − q21 ≤ −7
−2x1 − x2 − q12 ≤ −7, −2x1 − x2 − q22 ≤ −17
−x1 + 3x2 ≤ 21, 4x1 + 3x2 ≤ 45, x1 + 3x2 ≤ 27, 3x1 + x2 ≤ 30
x1, x2, γ1, γ2 ≥ 0


(23)

When the model above (23) is solved
using the MATLAB-R2023 program, γ1 =
γ2 = 0 is the outcome. This suggests that
the Pareto-optimal solution to IF-MOLDMP
(20) is provided by the found solutions
x1 = 6,x2 = 7. The suggested method, how-
ever, allows decision-makers to select the goal
that must be accomplished first in order of
importance. For instance, as δ ↗(increases),
the value of f1(x) ↗ and the value of
f2(x) ↘ (decreases), etc.

8. CONCLUSION

In this paper, the penalized intuition-
istic fuzzy goal programming strategy has
been proposed for finding Pareto-optimal so-
lutions to the IF-MOLDMP in an intuitionis-
tic fuzzy decision environment. When applied
to optimization problems in an imprecise en-
vironment, the IFO methodology is among
the most effective methods available, yield-
ing more satisfactory outcomes than fuzzy
and classical optimizations. The signifi-
cant contribution of the proposed method is

the development of an extended intuition-
istic fuzzy interactive technique to deter-
mine the most preferred Pareto-optimal for
IF-MOLDMP in an intuitionistic fuzzy en-
vironment. This technique combines the
penalty function method with an appropri-
ate intuitionistic fuzzy aggregation operator.
When compared to the existing IFO method,
the proposed method can choose from a set
of compromise solutions that are both effi-
cient and meet the DM’s preference for IF-
MOLDMP. Furthermore, the advantages of
the proposed method are that there is no need
to add an extra zero-one variable, it elimi-
nates the limitation of overestimation or un-
derestimation of the nadir point when estab-
lishing a reference point for decision-makers,
and it guarantees that the existing solution
x∗ satisfies the Pareto-optimally condition.

A future study could focus on applying
the proposed solution method to different
real-world application problems, such as wa-
ter resource allocation and inventory control
problems, and comparing the outcomes to ex-
isting optimization methods.

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54



Acknowledgments

All authors are grateful to the editor-in-
chief and anonymous reviewers for provid-
ing in depth comments and suggestions that
enhanced the clarity and readability of the
manuscript.

Conflict of Interest

On behalf of all authors, the correspond-
ing author state there is no conflict of interest.

Funding

The authors did not receive any funding
for this research.

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57


	INTRODUCTION
	MATHEMATICAL MODEL FORMULATION
	The modified mathematical model
	Model formulation

	QUALITATIVE ANALYSIS OF THE MODIFIED MODEL
	Well-posedness
	Steady state
	Basic reproduction number
	Local stability of disease free equilibrium
	Global stability of disease free equilibrium
	Endemic equilibrium point
	Local stability of endemic equilibrium
	Bifurcation analysis
	Sensitivity analysis

	NUMERICAL SIMULATIONS AND DISCUSSION
	EXTENSION OF THE MODIFIED MODEL INTO AN OPTIMAL CONTROL
	Optimal protection and hospitalization using modified model
	Existence of an optimal control
	The Hamiltonian and optimality system

	Numerical simulations of optimal control problem
	Optimal control comparisons and strategies

	CONCLUSION
	INTRODUCTION
	PRELIMINARIES
	Intuitionistic Fuzzy Set
	Operations over Intuitionistic Fuzzy Sets

	Multi-objective programming problem

	MATHEMATICAL FORMULATION OF PROBLEM
	Intuitionistic Fuzzy Multi-objective Linear Decision-Making Problem
	Intuitionistic Fuzzy Goal Model of IF-MOLDMP

	PROPOSED SOLUTION METHOD
	 Extended Yager-membership function in the IFDE
	Develop a new intuitionistic fuzzy (IF) aggregation operator
	The intuitionistic fuzzy goal programming method
	Interactive Penalty Function Method (IPFM)

	ALGORITHM FOR PENALIZED IFGP METHOD
	NUMERICAL EXAMPLE
	RESULTS AND DISCUSSION
	CONCLUSION
	INTRODUCTION
	MATHEMATICAL MODEL FORMULATION
	The modified mathematical model
	Model formulation

	QUALITATIVE ANALYSIS OF THE MODIFIED MODEL
	Well-posedness
	Steady state
	Basic reproduction number
	Local stability of disease free equilibrium
	Global stability of disease free equilibrium
	Endemic equilibrium point
	Local stability of endemic equilibrium
	Bifurcation analysis
	Sensitivity analysis

	NUMERICAL SIMULATIONS AND DISCUSSION
	EXTENSION OF THE MODIFIED MODEL INTO AN OPTIMAL CONTROL
	Optimal protection and hospitalization using modified model
	Existence of an optimal control
	The Hamiltonian and optimality system

	Numerical simulations of optimal control problem
	Optimal control comparisons and strategies

	CONCLUSION

