Jurnal Teknik Industri Vol.
No.
August 2024, pp.
ISSN : 1978-1431 print | 2527-4112 online Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming Dinita Rahmalia a*.
Nadya Husenti b a Department of Mathematics.
University of Islamic Darul Ulum Lamongan.
Lamongan.
Indonesia b Department of Informatics.
University of Muhammadiyah Gresik.
Jalan Sumatra GKB Gresik.
Indonesia *Corresponding author: dinitarahmalia@gmail.
ARTICLE INFO
ABSTRACT
Article history Received.
December 27, 2023 Revised.
March 25, 2024 Accepted.
July 31, 2024 available online.
August 31, 2024 The increasing complexity of stock trading requires effective portfolio management to optimize returns while minimizing Portfolio selection is critical in determining the most suitable combination of stocks, aiming to maximize expected returns and minimize risk within a given investment limit.
This study constructs a mathematical model for portfolio optimization using six different stocks, incorporating constraints such as expected return, risk, and available investment.
Given the multiobjective nature of the problem, a hybrid approach is proposed, combining Compromise Programming (CP).
Nadir Compromise Programming (NCP), and Ant Colony Optimization (ACO) to address both minimization and maximization objectives.
The ACO algorithm is applied to minimize deviation variables, which serve as the fitness function in the optimization process.
The results demonstrate the effectiveness of the hybrid method in selecting portfolios that achieve minimal deviation, providing an optimal balance between risk and return.
This research offers valuable insights for investors by illustrating the trade-offs between risk and reward in stock selection, contributing to more informed decision-making in portfolio management.
Keywords Portfolio Compromise Programming Ant Colony Optimization Multi-Objective Metaheuristic This is an open-access article under the CCAeBY-SA license.
Introduction Investors generally seek to create portfolios that yield long-term benefits.
Stocks, one of the most common assets, fluctuate in prices influenced by market demand and supply over time.
Most investors aim to construct portfolios that maximize expected returns while minimizing risk within the constraints of available capital.
Given the volatility in stock prices, portfolio selection becomes crucial to balance these objectives.
Statistical measures such as return, expected return, and stock risk can be computed based on historical data, providing valuable insights for decision-making .
Various studies have explored portfolio models and their modifications.
For example, models focusing on portfolio stability and minimizing risk have been developed .
Fuzzy preference techniques have been applied to portfolio selection .
, while other studies have constructed efficient portfolios with fewer stocks .
Furthermore, development https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
http://ejournal.
id/index.
php/industri jurnal@umm.
Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
ISSN : 1978-1431 print | 2527-4112 online Jurnal Teknik Industri Vol.
No.
August 2024, pp.
costs associated with portfolio selection have been suggested for decision-makers .
Predicting expected returns and risk involves forecasting future stock values .
, and synergies between projects have been shown to impact portfolio decisions .
Extendable investments have been incorporated into portfolio models .
, and pre-selected assets have been used for portfolio optimization .
Additionally, multi-objective portfolio selection models have been proposed, considering variable risk .
and return distributions .
Several previous studies have utilized different methods to estimate stock prices.
For example, the Kalman Filter .
and H-infinity .
methods rely on predictor and corrector iterations to make estimations.
Other approaches include Neural Networks .
and Adaptive Neuro-Fuzzy systems .
, which train data and then apply the optimized parameters during testing with a set data proportion.
The Autoregressive Integrated Moving Average (ARIMA) model .
, leveraging autocorrelation, has also been widely used.
All these methods aim to minimize the error between actual stock prices and forecasted data.
In this research, we develop a portfolio optimization model using Compromise Programming, which is well-suited for solving multi-objective problems by finding a compromise solution that balances two or more conflicting objectives .
The basic principles of Compromise Programming have been applied to a range of problems, including general portfolio selection .
, resource allocation .
, multi-objective shipment problems .
, multi-objective task assignment .
, and energy generation planning .
Ant Colony Optimization (ACO) is an optimization method inspired by the behavior of ants in searching for food and navigating their environment using This method, developed by Dorigo in the 1990s, simulates how ants traverse through various nodes from their nest to a food source.
ACO has been extensively researched and has proven to optimize search paths and resource allocation efficiently .
Additionally.
ACO has been successfully combined with other algorithms such as Genetic Algorithm (GA) .
Variable Neighborhood Descent .
, and Simulated Annealing .
to improve outcomes.
Applications of ACO span diverse areas, including traffic management systems .
, resource optimization .
, completion time minimization .
, vehicle routing problems .
, distribution planning .
, and open shop scheduling problems .
Previous studies have focused mainly on applying Ant Colony Optimization (ACO) to single-objective optimization problems, either addressing only minimization or maximization, which limits its applicability to more complex, real-world scenarios.
Similarly.
Compromise Programming has traditionally been solved through analytical methods, resulting in inefficient computations for large-scale problems.
These limitations highlight the need for more robust and efficient methods capable of handling multiple objectives simultaneously, particularly in portfolio optimization, where both risk minimization and return maximization are critical.
This research introduces a novel hybrid approach combining ACO with Compromise Programming and Nadir Compromise Programming to address these gaps.
Given investment constraints, this study aims to develop an optimization model that minimizes portfolio risk while maximizing expected returns.
By integrating these methods, the study seeks to overcome the limitations of previous research and provide a more efficient solution to the multiobjective optimization problem in portfolio management.
The contributions of this research are twofold: .
Practically, it offers investors valuable insights into managing the trade-offs between risk and return when selecting stocks, thus enhancing decisionmaking in portfolio construction.
Theoretically, it demonstrates the potential of Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
Jurnal Teknik Industri Vol.
No.
August 2024, pp.
ISSN : 1978-1431 print | 2527-4112 online ACO as a metaheuristic that can be effectively applied to multi-objective optimization problems, broadening its scope and applicability beyond single-objective cases.
Methods This study begins with the statistical computation of expected return and risk for a set of stocks .
Before determining the expected return, we calculate each stock's return per time unit, as shown in Equation .
If the return is positive, the stock generates a profit.
otherwise, it results in a loss.
Rit = St Oe St Oe1 St Oe1 Where:
: i-th return of stock when time t Rit St Oe1 : the price of the stock when the time t : the price of stock when time t Oe 1 The expected return for stock ycn over the period is calculated as the mean return, or average return, over time.
It is expressed in Equation .
A positive expected return indicates profitability, while a negative value indicates a loss.
EuR E ( Ri ) = t =1 .
Where T is the number of periods.
To compute the risk i , we calculate the covariance between the stock's and the market returns, represented by the Indonesia Composite Index (ICI).
The formula for risk is given in Equation .
, and the covariance is used to measure how changes in one variable are related to changes in another variable.
Bi = cov( Ri .
Rh ) A h2 The formula of covariance to stock can be seen in Equation .
, and market variance are in Equation .
, respectively.
Return Rht is the market return represented by ICI.
Eu ( R Oe E ( R ) )( R Oe E ( R ) ) cov( Ri .
Rh ) = t =1 .
Eu ( R Oe E(R )) A h2 = t =1 .
Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
ISSN : 1978-1431 print | 2527-4112 online Jurnal Teknik Industri Vol.
No.
August 2024, pp.
The methods employed in this research involve a hybrid approach combining Ant Colony Optimization (ACO) with Compromise Programming (CP).
A hybrid of ACO and Nadir Compromise Programming (NCP) is also used.
The process begins by generating a set of feasible solutions, i.
, the proportion of the selected portfolio and determining the number of ants.
The pheromone parameters are initialized uniformly.
During each iteration, the pheromone values are updated based on the deviation as the objective value, with each ant selecting candidate solutions based on the updated pheromone levels.
1 Compromise Programming Zeleny introduced the Compromise Programming (CP) method in 1974 .
Compromise Programming can solve multi-objective problems by finding the best compromise solution in optimizing two or more objectives.
In Compromise Programming, we optimize fi , minimize f j , and maximize f k then they can be written in Equation .
Opt fi .
= 1, 2,3,.
N A )
min f j ( j = 1, 2,3,.
N B )
max f k .
= 1, 2,3,.
NC )
Overall, the CP model for optimizing fi , minimizing f j , and maximizing f k as follows:
E NA
min E Eu wi (A i A iOe ) p Eu w j (A j ) p Eu wk (A kOe ) p E j =1 k =1 E i =1 E Subject to in Equation .
fi A iOe A i = f .
) , i = 1, 2,.
N A
, j = 1, 2,.
N B
Oe , k = 1, 2,.
A .
A .
A .
A , wi , w j , wk C 0 f j OeA = f fk A = f Oe Oe Euw Euw Euw =1 i =1 j =1 k =1 .
where the f.
) is the target function of i-th objective, f j is the ideal minimum of j-th objective function, f kmax is the ideal maximum of yco Oe ycEa objective function.
A Oe is the negative deviation and A is the positive deviation.
The mathematical formulation can be seen in Equation .
when applied to the portfolio optimization model.
min Z = (A1Oe A1 ) (A 2Oe ) (A 3 ) .
Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
Jurnal Teknik Industri Vol.
No.
August 2024, pp.
ISSN : 1978-1431 print | 2527-4112 online Subject to :
Eu x A OeA = N Oe i =1 .
Eu E(R ) x A = N R Eu B x OeA = N Z xi C 0 i =1 i =1 i = 1, 2,.
, n A .
A C 0 j = 1, 2,3 Oe With decision variables and parameters :
xi , i = 1, 2,.
, n is the proportion of selected portfolio as decision variable : total investment E ( Ri ) : expected return of i-th stock : the risk of i-th stock : risk of portfolio Equation .
represents the minimization of deviation variables derived from constraints .
Constraint .
ensures that the total proportion of the selected portfolio equals the available investment.
Constraint .
requires that the expected return of all selected stocks be greater than the average return, incorporating a deviation variable yu2Oe for maximization.
Lastly, constraint .
limits the total portfolio risk to be less than or equal to a predefined risk threshold, using the deviation variable Oeyu3 for minimization.
2 Nadir Compromise Programming Nadir Compromise Programming (NCP) is an extension of the Compromise Programming (CP) method, introduced in 2011.
NCP is designed to address multiobjective optimization problems by simultaneously optimizing, minimizing, and maximizing objectives.
This approach modifies the original CP framework to improve performance in specific contexts .
The general NCP model is formulated as Equation .
E NA
min E Eu wi (A i A iOe ) p Eu w j (OeA jOe ) p Eu wk (OeA k ) p E j =1 k =1 E i =1 E Subject to :
fi A iOe A i = f .
) , i = 1, 2,.
N A
, j = 1, 2,.
N B
Oe , k = 1, 2,.
A .
A .
A .
A , wi , w j , wk C 0 f j A = f Oe fk Oe A = f Oe Euw Euw Euw =1 i =1 j =1 k =1 .
Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
ISSN : 1978-1431 print | 2527-4112 online Jurnal Teknik Industri Vol.
No.
August 2024, pp.
When applied to portfolio optimization, the NCP model can be seen in Equation .
min Z = (A1Oe A1 ) Oe (A 2 ) Oe (A 3Oe ) .
Subject to :
Eu x A OeA = M i =1 Oe Eu E(R ) x Oe A = M R EuA x A = M S .
xi C 0 i =1 i =1 Oe i = 1, 2,.
, n A .
A C 0 j = 1, 2,3 Oe Equation .
represents the minimization of slack and surplus variables derived from the constraints in Equations .
Constraint .
ensures that the total proportion of selected stocks equals the available investment.
Constraint .
requires that the expected return of all selected stocks exceed the average return, with the slack variable Oeyu2 addressing the maximization requirement.
Constraint .
ensures that the overall portfolio risk remains below a certain threshold, with the surplus variable yu3Oe managing the minimization aspect.
Ant Colony Optimization Ant Colony Optimization (ACO) is an algorithm inspired by the behavior of ants in their search for food and nesting sites.
In this algorithm, ants depart from the nest and traverse through multiple nodes, starting from the first layer .
to the last layer .
, ultimately stopping at their destination .
This method was introduced by Dorigo in 1990.
For the portfolio selection model, the ACO algorithm can be structured as follows:
Set the number of ants N and the pheromone decay factor A .
Generate P feasible solutions i.
the proportion of selected portfolio X k , k = 1, 2,.
with the design X = xi , i = 1, 2,.
, n with n is the number of stocks.
In generating population, there are some constrains that should be satisfied like Eu x = M , so that initialization of feasible solutions can be constructed as follows i =1 for k = 1: P while ( p == .
q = rand .
if( q C 0.
take two stocks randomly take three stocks randomly Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
Jurnal Teknik Industri Vol.
No.
August 2024, pp.
ISSN : 1978-1431 print | 2527-4112 online xi C n i , i = 1, 2,.
, n Eu xi i =1 Compute A1Oe .
A1 Compute A 2 = M 0 R Oe Eu E ( Ri ) xi i =1 Compute A 3Oe = Eu Ai xi Oe M 0 S i =1 if( A .
A .
A .
A C 0 ) p =1 Oe Oe Give the uniform probability.
, k = 1, 2,.
Calculate cumulative probability range C k k = 1, 2,.
p( X k ) = .
Generate random variable rs U .
s = 1,2,.
N .
Determine selected variable X k , k Ea.
, 2,.
P} for every ant s .
Calculate objective function f ( X k ) for every ant s .
Choose minimum fitness function fbest = min ( f ( X k ), k Ea .
, 2,.
P}) , and count N best , the number of f best Set constant Q and calculate Eu AEA ( X k ) , k = 1, 2,.
ycE , if ycUyco is the best variable yceycayceycyc Update the pheromone based on Equation .
A k = .
Oe A )A k Eu AEA ( X k ) , k = 1, 2,.
ycA .
Oc OIyua.
cUyco ) = { ycayceycyc Update the pheromone probability based on Equation .
A p( X k ) = k , k = 1, 2,.
EuA k Repeat step 3-10 until all ants choose the best path consisting pheromone and process converges.
Data The data used in the experiments for the hybrid Compromise Programming and Nadir Compromise Programming models are obtained from six stock datasets covering the period from January 2016 to December 2018.
The stocks analyzed include Kimia Farma (KAEF).
Telekomunikasi Indonesia (TLKM).
Gudang Garam (GGRM).
Matahari Department Store (LPPF).
Garuda Indonesia (GIAA), and Bank Central Asia (BBCA).
For each stock, the expected return is computed using Equation .
based on return data over the selected period, while the risk is calculated using Equation .
The results of these computations are presented in Table 1.
Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
ISSN : 1978-1431 print | 2527-4112 online Jurnal Teknik Industri Vol.
No.
August 2024, pp.
Table 1.
Expected return and risk of each stock
Stock
Expected Return
KAEF
TLKM
GGRM
LPPF
GIAA
BBCA
Risk With total investment N0 = 1 , risk Z = 0.
9 , and average of expected return R = 0.
In Ant Colony Optimization, parameters used both in Compromise Programming and Nadir Compromise Programming are :
The number of ants : 10.
20, 30
Maximum iterations : 25, 50 100 Results and Discussion 1 Simulation Result of Compromise Programming After calculating each stock's expected return and risk, the Ant Colony Optimization (ACO) algorithm was constructed using the earlier parameters.
In each iteration, ants randomly select candidate solutions.
The best solution from all ants is then identified, and the pheromone level for this solution is increased, improving its likelihood of being selected in subsequent iterations.
The results of the simulation for the hybrid Ant Colony Optimization and Compromise Programming are shown in Figure 1.
Figure 1.
Simulation result of hybrid Ant Colony Optimization and Compromise Programming The simulation shows that ants select candidate solutions randomly from the feasible set in the initial iteration, as the pheromone probability is evenly distributed.
Once the best solution is identified, the pheromone probability for this solution is updated, increasing its chances of being selected in the next iteration.
As the iterations progress, the algorithm converges, and after reaching the maximum iteration, the optimal investment proportions for each stock are determined.
Table 2 shows that KAEF Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
Jurnal Teknik Industri Vol.
No.
August 2024, pp.
ISSN : 1978-1431 print | 2527-4112 online and GIAA stocks are selected for investment, with proportions of 23.
65% and 76.
KAEF
Table 2.
Investment proportion for each stock
TLKM
GGRM
LPPF
GGIA
BBCA
Based on the sum of deviation variables, the resulting fitness value is 0.
We further extended the experiment by varying the number of ants and iterations.
Table 3 summarizes the results under different configurations.
It can be seen that GGRM and GIAA stocks are frequently selected across different scenarios due to their relatively low risk, as indicated in Table 1, where their risk values are both less than 1.
The simulation results demonstrate that the hybrid ACO and Compromise Programming method can optimize portfolio selection by identifying stocks with favorable risk-return profiles.
In particular, stocks like GGRM and GIAA exhibit lower They are frequently selected across different iterations and ant configurations, indicating their robustness in various scenarios.
This research provides valuable insights for investors seeking to balance risk and return in their portfolios.
The hybrid approach offers a systematic way to minimize risk while maximizing returns, leading to more informed investment decisions.
Additionally, the flexibility of the ACO algorithm in selecting optimal solutions based on pheromone probabilities highlights its potential in complex multi-objective optimization problems.
The findings suggest that ACO, combined with Compromise Programming, can significantly improve portfolio optimization processes, offering theoretical contributions to optimization methods and practical implications for investment strategies.
Table 3.
ACO on Compromise Programming with Different Numbers of Ants and Iterations Total
Ant
Maximum
Iteration KAEF
TLKM
GGRM
LPPF
GGIA
BBCA
Fitness 2 Simulation Result of Nadir Compromise Programming The computation process for the hybrid Ant Colony Optimization (ACO) and Nadir Compromise Programming (NCP) is similar to that of Compromise Programming, with the primary difference being the deviation variables used.
After computing each stock's expected return and risk, the ACO algorithm is constructed based on the defined In each iteration, ants randomly select candidate solutions.
The best solution from all ants is then selected, and the pheromone levels for that solution are updated to increase its likelihood of being chosen in the subsequent iterations.
The Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
ISSN : 1978-1431 print | 2527-4112 online Jurnal Teknik Industri Vol.
No.
August 2024, pp.
simulation results for the hybrid ACO and Nadir Compromise Programming are shown in Figure 2.
Figure 2.
Simulation result of hybrid Ant Colony Optimization and Nadir Compromise Programming In the initial iteration, ants select candidate solutions randomly due to the uniform pheromone distribution across all options.
Once the best solution is identified, the probability of selecting that solution increases in subsequent iterations due to the pheromone update mechanism.
The optimal investment proportions for each stock are determined upon reaching the maximum iteration.
Table 4 shows that KAEF and TLKM stocks are selected for investment, with proportions of 10.
88% and 89.
12%, respectively.
Table 4.
Investment proportion for each stock
KAEF
TLKM
GGRM
LPPF
GGIA
BBCA
Based on the sum of deviation variables, the resulting fitness value is -0.
Furthermore, we extended the experiment by varying the number of ants and iterations.
Table 5 summarizes the results for different configurations.
From the table, it can be observed that stocks KAEF and TLKM are frequently selected across different ant and iteration settings.
This is because, as shown in Table 1.
KAEF has the highest expected return, while TLKM exhibits the lowest risk.
Table 5.
ACO on Nadir Compromise Programming with Different Numbers of Ants and Iterations Total
Ant
Maximum
Iteration KAEF
TLKM
GGRM
LPPF
GGIA
BBCA
Fitness Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
Jurnal Teknik Industri Vol.
No.
August 2024, pp.
ISSN : 1978-1431 print | 2527-4112 online The results of the Nadir Compromise Programming simulations indicate that the stocks KAEF and TLKM consistently emerge as the preferred investment options across various iterations and ant configurations.
It is because KAEF has the highest expected return, making it an attractive option for maximizing profit.
At the same time.
TLKM exhibits the lowest risk, making it a stable choice for risk-averse investors.
These findings demonstrate the effectiveness of the hybrid ACO and Nadir Compromise Programming approach in portfolio optimization, providing a robust method for balancing risk and return.
The flexibility of this method allows for efficient exploration of multi-objective optimization problems, making it a valuable tool for investors seeking to construct well-balanced portfolios.
Moreover, fine-tuning the number of ants and iterations offers additional control over the optimization process, ensuring that the results can be adapted to different investment scenarios.
Conclusion This study demonstrates that the optimization models of Compromise Programming and Nadir Compromise Programming can effectively assist investors in determining the optimal portfolio composition, considering constraints such as investment amount, expected return, and risk.
The critical contribution of this research is integrating the Ant Colony Optimization (ACO) algorithm with both Compromise Programming and Nadir Compromise Programming.
Inspired by ants' behavior in searching for food and building nests through pheromone-based communication.
ACO was used to explore feasible solutions for portfolio selection.
As the iterations progress, ants refine their choices based on pheromone levels, leading to an optimal solution.
The results indicate that Nadir Compromise Programming outperforms Compromise Programming by consistently selecting stocks with the highest expected return and the lowest risk.
This makes it a more robust method for portfolio optimization.
The approach minimizes deviation variables, resulting in a highly efficient fitness function that converges to the best portfolio configuration.
However, the study has some limitations.
The model relies on historical stock data and does not account for potential future market changes or external factors that might influence stock performance.
Additionally, the fixed weight assignment for portfolio components may limit the model's flexibility in handling more dynamic market For future research, exploring a fuzzy approach to determine the weight of each portfolio component in both Compromise Programming and Nadir Compromise Programming is recommended.
This would allow for a more adaptable model to accommodate uncertainty and variability in market conditions better.
Declarations Author contribution: We declare that all authors contributed equally to this paper and approved the final paper.
Funding statement: No funding was received for this work.
Conflict of interest: The authors declare no conflict of interest.
Additional information: No additional information is available for this paper.
Please cite this article as: Rahmalia.
, & Husenti.
Multi-Objective Portfolio Optimization Using Hybrid Ant Colony Optimization and Compromise Programming .
Jurnal Teknik Industri, 25.
, 131Ae144.
https://doi.
org/10.
22219/JTIUMM.
Vol25.
No2.
ISSN : 1978-1431 print | 2527-4112 online Jurnal Teknik Industri Vol.
No.
August 2024, pp.
Acknowledgments Author would like appreciate to the Ministry of Research and High Education (Kemenristekdikt.
, the Republic of Indonesia for the Penelitian Dosen Pemula 2023
References