IMPLEMENTATION OF FUZZY MODEL TAHANI IN DECISION SUPPORT SYSTEM FOR OPTIMAL PRODUCTION SCHEDULING

Rizaldi1, Arridha Zikra Syah2*, Dewi Anggraeni3 
Informastion System, Faculty of Computer Science, Royal University
email: azsyra@gmail.com

Abstract: In the manufacturing industry, production scheduling become an important aspect that affects operational efficiency and customer satisfaction. The main challenge in scheduling is optimizing the use of resources to meet demand by minimizing production costs and time. Suboptimal scheduling can lead to problems such as delays in stocking, stock buildup, and increased operational costs. Thus, a method can to handle the complexity and uncertainty in the production process is needed. The Fuzzy Tahani Model is an approach in decision support systems. this can be used to help companies achieve more efficient and adaptive production scheduling, to consider various variables such as demand, production capacity, and inventory levels. This research aims to develop and implement the model in the context of production scheduling, with the hope of improving operational performance and customer satisfaction. At this time, the proposed Fuzzy Model Tahani technology is in TKT 4, which is the validation stage of technology components in a laboratory environment. This research will advance technology development until it reaches TKT 6, at which point the model will be rigorously tested in a complex environment that closely simulates real industrial conditions. It is hoped that the findings of this research will be directly applied in the industrial production process to address current scheduling challenges.
Keywords: Fuzzy Model Hold, Production Scheduling, Decision Support System; Production Optimization 

Abstrak: Dalam industri manufaktur, penjadwalan produksi adalah aspek penting yang mempengaruhi efisiensi operasional dan kepuasan pelanggan. Tantangan utama dalam penjadwalan adalah mengoptimalkan penggunaan sumber daya untuk memenuhi permintaan dengan meminimalkan biaya dan waktu produksi. Penjadwalan yang tidak optimal dapat menyebabkan masalah seperti keterlambatan pengiriman, penumpukan stok, dan peningkatan biaya operasional. Oleh karena itu, diperlukan suatu metode yang mampu menangani kompleksitas dan ketidakpastian dalam proses produksi. Fuzzy Model Tahani adalah salah satu pendekatan yang dapat digunakan dalam sistem pendukung keputusan untuk membantu perusahaan mencapai penjadwalan produksi yang lebih efisien dan adaptif, dengan mempertimbangkan berbagai variabel seperti permintaan, kapasitas produksi, dan tingkat persediaan. Penelitian ini bertujuan untuk mengembangkan dan mengimplementasikan model tersebut dalam konteks penjadwalan produksi, dengan harapan dapat meningkatkan performa operasional dan kepuasan pelanggan. Pada saat ini, teknologi Fuzzy Model Tahani yang diusulkan berada pada TKT 4, yaitu tahap validasi komponen teknologi dalam lingkungan laboratorium. Penelitian ini akan melanjutkan pengembangan teknologi hingga mencapai TKT 6, di mana model yang dikembangkan akan diuji dalam lingkungan yang lebih kompleks dan mendekati kondisi industri nyata. Dengan demikian, diharapkan hasil penelitian ini dapat diaplikasikan secara langsung dalam proses produksi industri untuk mengatasi tantangan penjadwalan yang ada.

Kata kunci: Fuzzy Model Tahani; Penjadwalan Produksi; Sistem Pendukung keputusan; Optimasi Produksi

INTRODUCTION

In the manufacturing industry, production scheduling is a critical factor in achieving operational efficiency and ensuring customer satisfaction. Poor scheduling can result in numerous issues, including delivery delays, excessive inventory buildup, and rising operational costs [1], [2].. As global competition intensifies, manufacturing companies must respond swiftly and efficiently to market demand.
Metode tradisional yang sering digunakan dalam penjadwalan produksi, seperti algoritma heuristik atau pendekatan linear programming, sering kali tidak mampu menangani kompleksitas dan ketidakpastian dalam lingkungan produksi [3]. Sistem ini digunakan untuk membantu pemecahan masalah yang bersifat semi-terstruktur atau tidak terstruktur, terutama dalam situasi yang memerlukan keputusan cepat dan tepat [4], [5]. Di sinilah sistem pendukung keputusan (SPK) berbasis Fuzzy Model Tahani menjadi relevan [6]. Fuzzy Model Tahani memungkinkan pengambilan keputusan dalam kondisi ketidakpastian dan dapat menangani data yang tidak eksak, sehingga menghasilkan penjadwalan produksi yang lebih optimal [7], [8].
Traditional methods commonly employed in production scheduling, such as heuristic algorithms and linear programming approaches, often struggle to manage the complexity and uncertainty present in production environments [3]. These techniques typically fall short when addressing semi-structured or unstructured problems, particularly in scenarios that demand quick and accurate decision-making. This is where the Decision Support System (DSS) based on the Fuzzy Tahani Model becomes significant [4], [5]. The Fuzzy Tahani Model facilitates decision-making under uncertain conditions and is capable of working with imprecise data, ultimately leading to improved production performance[7], [8].
According to data from the International Journal of Production Research (2023), approximately 60% of manufacturing companies in Southeast Asia report issues with inefficient production scheduling. This problem is also experienced by Mr. Ediyal. Furniture makers face challenges in managing their time for tasks related to frames, doors, windows, and other components, both on a weekly and monthly basis. Additionally, the demand for different models and product sizes varies among consumers. This situation highlights the need for more adaptive and responsive methods, such as the Fuzzy Model Tahani, to effectively address these challenges.
This research aims to develop and implement a decision support system based on the Fuzzy Model Tahani to optimize production scheduling in manufacturing companies. The system is expected to improve operational efficiency and reduce production costs by minimizing uncertainties in the scheduling process.
In the calculation of the Tahani Model using the Fuzzy method, the variables involved are categorized into input, process, and output variables, all defined by fuzzy membership functions. 
The input variables (independent) include Demand, Production Capacity, and Inventory Level. The process variables (intermediate) consist of fuzzy membership functions and fuzzy rules. Finally, the output variable (dependent) is the Production Schedule.
The technology addressed in this study is firmly positioned at the Technology Wastefulness Level (TKT) 4-5. At this level, the Fuzzy Model Tahani method has been effectively implemented on a laboratory scale or during initial trials; however, its application in the industry remains limited. This research is set to elevate the TKT to levels 6-7, where a robust Fuzzy Model-based decision support system will be implemented in real production environments, accompanied by thorough evaluations of its performance.

METHOD

1.	Fuzzy Inference System (FIS)
Lotfi A. Zadeh introduced fuzzy sets as an extension of classical set theory [9]. In a classical set, the membership of an element is indicated by either 0 or 1. A value of 1 signifies that the element is a member of the set, while a value of 0 indicates that the element is not a member of the set. In a fuzzy set, the membership of an element is clearly expressed by a value between 0 and 1, known as the membership function (μA(x)). This function unequivocally defines the degree to which element x belongs to fuzzy set A. This capability effectively represents uncertainty and ambiguity in data and information. [6], [10].
The function of fuzzy logic can be calculated using the following formula:

μA(x):X→[0,1]		(1)

Let X represent the set of universes, and μA(x) denote the degree of membership of element x in fuzzy set A.
Membership functions are fundamental to fuzzy sets, and their shapes can be effectively customized to suit the specific requirements of various applications. The following are some of the most commonly utilized membership functions:
a.	Triangular Membership Function
	(2)
Where a, b, and c are the parameters that determine the shape of the triangle. This function is simple and is often used for linear representations.
 
     1 |             /\
        |            /  \
        |          /      \
     0 |____/_____\__
         2   5            8
Image 1. Triangle function chart

In this graph, the value x=2 has a membership of 0 (non-member), x=5 has a membership of 1 (full member), and x=8 again has a membership of 0.

b.	Trapezoidal Membership Function
		(3)
This function is more flexible than the triangle function because it allows for a flat area of peaks, reflecting greater uncertainty.

     1 |              ______
        |            /              \
        |          /                    \
     0 |____/_____________\__
         1   3                6           9
Image 2. Trapezoidal Membership Function

In this graph, a value of x between 3 and 6 has full membership (1), while a value of x below 1 or above 9 has no membership (0).

c.	Gaussian Membership Function
	(4)
Where c is the center of the Gaussian function and σ is the width of the curve. This function is used for more subtle and frequent representation in statistical analysis.
     1 |        __/^\__
        |         /        \
        |       /            \
     0 |__/_________\____
        0   2  5     7   10
Image 3. Gaussian Membership Function Graph

In this graph, the value x=5 has full membership (1), while the surrounding values decrease exponentially from the center.

d.	Operation of the Fuzzy Assembly
Zadeh introduced the concepts of Fuzzy Operations, including Union (∪), Intersection (∩), and Complement (¬), which are defined as Union, Insertion, and complement.
The union operation states the highest degree of membership between the two fuzzy sets A and B for element x. Union operations are defined by formula (5).

μA∪B(x)=max(μA(x),μB(x))	(5)

The intersection operation describes the lowest degree of membership between fuzzy sets A and B for element x. Insertion operations are defined by the following formula:

μA∩B(x)=min(μA(x),μB(x))	(6)

The complement operation reveals how much element x belongs to the fuzzy set A. Complement operations are defined by the formula (7).

μ¬A(x)=1−μA(x)		(7)

e.	Fuzzy Relation 
Fuzzy relations extend the concept of relations in classical set theory. Let X and Y be two sets. A fuzzy relation R between X and Y is represented as a fuzzy set within the Cartesian product X × Y. This fuzzy relation is defined by a membership function that assigns a degree of membership for each element in the Cartesian product.

μR(x,y):X×Y→[0,1]	(8)

Fuzzy relations model the connection between two fuzzy sets, particularly in control and decision-making systems.

2.	Fuzzy Tahani Model
The Fuzzy Tahani model is a method used in Decision Support Systems (DSS) that employs fuzzy sets to address uncertainty and ambiguity in decision-making processes. Developed by Jamshid Tahani in 1977, this model has found applications in various fields, including production management, project selection, and performance appraisal.
The Basic Principles of the Fuzzy Tahani Model effectively combine fuzzy logic with relational databases. This model utilizes fuzzy sets to accurately represent ambiguous or uncertain information. It employs a fuzzy pattern matching mechanism to draw robust conclusions and to query the fuzzy database confidently [11].
The main components of the Fuzzy Tahani Model include:
a.	Fuzzy Set
Every attribute in a database must be represented by a fuzzy set. For example, the "Temperature" attribute will have fuzzy sets like "Cool," "Medium," and "Hot," each defined by a specific membership function. This approach ensures precise categorization and analysis of data.
b.	Fuzzy relations
Fuzzy relations replace traditional relational models in databases. This relationship not only connects two tables or attributes but also considers the degree of fuzzy membership of the involved elements.
c.	Fuzzy Query
Users can submit queries that include vague terms. For instance, the query "Search for all products with high quality and low price" will utilize a fuzzy membership function to interpret what is meant by "high quality" and "low price."
d.	Inference Algorithm
This algorithm is designed to match the data in the database with the fuzzy conditions specified in the query. It evaluates each entity based on its degree of membership in the relevant fuzzy set.
The stages of the implementation of the Fuzzy Tahani Model are in the form of:
a.	Defining Fuzzy Sets for Attributes
Define a fuzzy set for each relevant attribute in the database. For example, for the "Temperature" attribute, the fuzzy set can be {Cold, Medium, Hot} with each membership function.
b.	Establishing Fuzzy Relations
Establish a clear fuzzy relationship to connect entities within a database. For example, define the relationship between "Product" and "Price" using fuzzy logic to assign each product a specific price membership degree of "Low," "Medium," or "High." This approach ensures a more nuanced understanding of pricing for each product.
c.	Fuzzy Query Process
Create a query that searches the database using fuzzy terminology. This process involves matching membership degrees and utilizing fuzzy operators such as "AND," "OR," and "NOT."
d.	Evaluation and Inference
Query evaluation using a fuzzy inference algorithm results in a list of entities that meet the defined fuzzy criteria, along with their degree of conformity.

RESULT AND DISCUSSION
A.	Result
1.	Variables
In the calculation using the Fuzzy Model Tahani method, the variables include input, process, and output variables, all defined by the fuzzy membership function. Below is an explanation of these variables.
a.	Input variables (independent variables)
1)	Demand
It measures the number of products requested by customers in a given period. Fuzzy Categories: Low, Medium, High.
2)	Production Capacity
This defines the maximum production capacity of the facility within that period. The capacity is categorized as follows: Low, Medium, High.
3)	Inventory Level
It indicates the quantity of product stock available in the warehouse before production begins. Fuzzy Categories: Low, Medium, High.
b.	Process Variables (Intermediate Variables)
1)	Fuzzy Membership Functions  
A function that specifies the degree of membership for each input, for example, how large an input value such as demand or production capacity is in the low, medium, or high categories.
2)	Fuzzy Rules  
Rules that exchange input for output are essential components of an inference system. This rule decisively determines the output based on specific input conditions. For instance, if demand is high and production capacity is high, then the production schedule must be high.
c.	Output Variable (Dependent variable)
The output variable is the Production Schedule, which determines the exact number of units to be produced in a specified timeframe. It uses fuzzy categories: Low, Medium, and High. The output is calculated with a defuzzification method, like the centroid method, to ensure accuracy.

2.	Fuzzy Inference Systems
Step 1: Definition of Input Variables
The three main input variables include Demand, Production Capacity, and Inventory Level. Each of these variables is associated with three fuzzy sets: high, medium, and low.
Step 2: Fuzzy Set Definition and Membership Function
Each input variable is represented by a fuzzy set, each with its own membership functions:
1)	Demand
Low: μ_low(d) = 1 if d ≤ 50, 0 if d ≥ 100, and linear between 50-100. Medium (Medium): μ_medium(d) = 0 if d ≤ 50 or d ≥ 150, linear between 50-100 and 100-150. High: μ_high(d) = 1 if d ≥ 150, 0 if d ≤ 100, and linear between 100-150.
2)	Production Capacity
Low: μ_low(c) = 1 if c ≤ 30, 0 if c ≥ 70, and linear between 30-70. Medium: μ_medium(c) = 0 if c ≤ 30 or c ≥ 100, linear between 30-70 and 70-100. High: μ_high(c) = 1 if c ≥ 100, 0 if c ≤ 70, and linear between 70-100.
3)	Inventory Level
Low: μ_low(i) = 1 if i ≤ 20, 0 if i ≥ 60, and linear anta-ra 20-60. Medium (Medium): μ_medium(i) = 0 if i ≤ 20 or i ≥ 80, linear between 20-60 and 60-80. High: μ_high(i) = 1 if i ≥ 80, 0 if i ≤ 60, and linear between 60-80.

Step 3: Definition of Output Variables (Production Schedule)
The output of the system is the production schedule, which will also be represented by the fuzzy set of Production Schedule (Production Schedule): Low: μ_low(p) = 1 if p ≤ 30, 0 if p ≥ 70, and linear between 30-70; Medium (Medium): μ_medium(p) = 0 if p ≤ 30 or p ≥ 100, linear between 30-70 and 70-100; High: μ_high(p) = 1 if p ≥ 100, 0 if p ≤ 70, and linear between 70-100.
Step 4: Developing Fuzzy Rules 
Here are some fuzzy rules for production scheduling: 1) If the demand is high, production capacity is high, and inventory levels are low, then the production schedule should be set to high; 2. If the demand is medium, production capacity is medium, and inventory levels are moderate, then the production schedule should be set to moderate; 3. If the demand is low, production capacity is low, and inventory levels are high, then the production schedule should be set to low.
Step 5: Fuzzy Inference
The data available is as follows: Demand is 120 units, production capacity is 80 units, and the inventory level is 50 units. 
Calculate the membership degree for each input variable and apply the fuzzy rules with the membership degree.

a.	Demand (for 120 units)
μ_low(120) = 0, This is below the minimum range); μ_medium(120) = (150-120)/(150-100) = 0.6; 
μ_high(120) = (120-100)/(150-100) = 0.4
b.	Production level(80 unit):
μ_low(80) = 0 , This is below the minimum range); μ_medium(80) = (100-80)/(100-70) = 0.67; μ_high(80) = (80-70)/(100-70) = 0.33
c.	Tingkat Persediaan (50 unit):
μ_low(50) = (60-50)/(60-20) = 0.25; μ_medium(50) = (80-50)/(80-60) = 0.5; μ_high(50) = 0, This is below the maximum range.
Application of Fuzzy Rules
a.	Rule 1: μ_high(120) * μ_high(80) * μ_low(50) = 0.4 * 0.33 * 0.25 = 0.033 (High Production Schedule)
b.	Rule 2: μ_medium(120) * μ_medium(80) * μ_medium(50) = 0.6 * 0.67 * 0.5 = 0.201 (Medium Production Schedule)
c.	Rule 3: μ_low(120) * μ_low(80) * μ_high(50) = 0 * 0 * 0 = 0 (Low Production Schedule).
Step 6: Defuzzification
After obtaining the fuzzy output, the next step is defuzzification, which provides a concrete value for the production schedule. The Centroid method is employed in this process of defuzzification. Assuming the output values for a high production schedule are set at 90 and for a medium production schedule at 60, the resulting production value can be effectively calculated as:

Step 7: Interpretation and Implementation  
The results of defuzzification yield a definitive production schedule value, which is approximately 64 units. This is the optimal production amount for the designated period (weekly/monthly).

B.	Discution
In this study, the Fuzzy Model Tahani method is used to solve the problem of production scheduling by considering three main input variables: De-mand, Production Capacity, and In-ventory Level. This process aims to determine an opti-mal production schedule by utilizing the concept of fuzzy sets and fuzzy rules that have been set. The following is a detailed discussion of the steps that have been taken.
In this initial step, we clearly define the input variables: Demand, Production Capacity, and Inventory Level. Each variable is categorized into three distinct levels: Low, Medium, and High. This structured approach effectively captures variations in input data and decisively informs our production schedules to align with the current conditions.
In Second stage, we establish fuzzy sets and corresponding membership functions for each input variable. These membership functions are essential tools that transform numerical input values into degrees of membership within various fuzzy categories. For instance, if the demand is recorded at 120 units, it may have a membership degree of 0.6 in the Medium category and a degree of 0.4 in the High category. This indicates that there is a moderate likelihood of the demand falling into the Medium category, along with some possibility in the High category. The membership functions we utilize are designed as linear functions, providing a smooth and flexible transition between different categories. This flexibility allows for a more nuanced understanding of how input values relate to fuzzy classifications.
Step Three is Definition of Output Variables (Production Schedule). The output of the system is the Production Schedule, represented as a fuzzy set comprising three categories: Low, Medium, and High. This fuzzy set allows for expressing the output in terms of membership degrees, which can capture variations in production schedule decisions more effectively.
The fourth step is crucial and involves the formation of fuzzy rules. These rules will clearly define the relationship between input variables and output outcomes. For instance, an effective rule states: "If demand is high, production capacity is high, and inventory levels are low, then the production schedule must be high." This rule embodies the decision-making logic based on input conditions and will play a vital role in the fuzzy inference process.
In the fifth step, fuzzy inference, we use the actual input data, which consists of a request for 120 units, a production capacity of 80 units, and a supply level of 50 units, to calculate the degree of membership for each input variable. Next, we apply fuzzy rules to produce fuzzy outputs. For instance, based on the existing rules, we determine that for the given combination of inputs, the membership degree for a High Production Schedule is 0.033, while for a Medium Production Schedule, it is 0.201.
After fuzzy inference, the next essential step is defuzzification, which transforms the fuzzy output into a precise value. We utilize the centroid method for this process, yielding a production schedule value of 64 units. This result is a direct outcome of combining the membership degrees of various output categories with the central values of the relevant fuzzy set.
The final step, Interpretation and Implementation, is crucial for achieving optimal results. The defuzzification results decisively indicate that the ideal production amount is 64 units for the specified period. This conclusion stems from a rigorous fuzzy analysis of the relevant input variables, including demand, production capacity, and current inventory levels.
Production managers must leverage these findings to effectively adjust production schedules, significantly reduce operational costs, and ensure product availability aligns precisely with market demands. By employing the Fuzzy Model Tahani method, the system confidently addresses uncertainty and variations in input data, resulting in superior operational efficiency in production scheduling.

CONCLUSION

Based on the findings from the research that utilized the Fuzzy Model Tahani method for completing the production scheduling timeframe, several conclusions can be drawn.
This study successfully determined that optimal production scheduling can be attained by taking into account the primary input variables: Demand, Production Capacity, and Inventory Level. By employing the Fuzzy Model Tahani method, the system is capable of generating production schedules that are more responsive to changes in input conditions, such as demand fluctuations and variations in production capacity.
Through applying the Fuzzy Model Tahani, this research has achieved this goal by providing a concrete and data-based solution for production scheduling. The results of the defuzzification show that the production schedule of about 64 units is an optimal number, according to the inputs given. This shows that the fuzzy method can provide accurate results for existing production needs.
The results show that the Fuzzy Model Tahani method can be used effectively to answer the formulation of this problem by considering various variables that affect production scheduling. The developed system is able to produce output in the form of an optimal production schedule based on fuzzy rules and defuzzification results that have been applied, so that it can be used as a tool in production decision-making.
Overall, this study has proven that the Fuzzy Model Tahani method is an effective and efficient tool in solving production scheduling problems, especially in complex and uncertain situations. This method is not only capable of overcoming variations in inputs but also provides a scalable and implementable solution in real industry scenarios.

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