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