Optimization of Teak Wood Furniture Production Using Linear Programming Method at Sumenep East Java Indonesia

Abstract


INTRODUCTION
In Kangean Island, the furniture company, namely 3R furniture, processed wood products into doors, chairs, cabinets, tables, and frames. The wood used as the primary material for production is teak wood. The company was founded on a land area of 900 M2 in 2005. The owner of this industry is Mr. Abdurrahman, who has pioneered and developed this industry until now. This company has some problems maximizing the benefit and minimizing the production cost. Determining the amount of production, a plan is needed. In furniture companies 3R, the capacity of production decisions still does not consider labor capacity and material availability.
For example, in the case of raw materials, the company cannot produce certain types because the raw materials are unavailable and are still waiting for delivery. Because the company uses a piece-rate system, the results obtained are different for each worker, both in terms of products produced and time. The availability of company resources at the 3R furniture company is a problem in production planning, which influences the expected profit and is hard to estimate. The 3R furniture company has not managed demand properly because it has not been able to synchronize the company's available resources with fulfilling consumer demand. The optimization of the production process cannot be achieved, and profits cannot be predicted correctly and precisely.
Production is the creation of products (goods/services). It also can be interpreted as an activity or process of transforming inputs into outputs. Operational management is a process or action of making products by transforming inputs into results. Production and operations management can also be defined as managing and coordinating various resources' effective and efficient use to create products or improve their value [1].
The Linear Programming method is an option that will determine the amount of production of each furniture product so that the optimization of the production process can be achieved precisely and adequately. Several previous studies have discussed the implementation of this linear programming [2], [3], [12], [13], [4]- [11].
The application of this linear programming also determines how many units must be produced by the 3R furniture industry located in Kangean. This furniture industry has several cabinets with the same primary raw material: teak wood. The products made include Decorative Cabinets, Tulet Hanging Cabinets, Chairs, Doors, Windows, Frames, and Tables. These seven products are the most often ordered from customers in large or small quantities. Based on the problems above, it is necessary to research to optimize the production process of furniture products to obtain maximum profit.

METHOD
In this study, data analysis uses a linear programming model simplex method. The simplex method is one of the solutions of linear programming in which the process of finding the solution is by using an iteration path, namely determining the possible point of the goal to be achieved with the help of a table until the optimal solution is obtained. The simplex method begins with one by one feasible point test to determine whether the objective function has achieved optimal results or has not achieved optimal results. When the results obtained from one possible point have not reached optimal results, proceed with the next feasible point, and so on until the objective function accepts optimal results if there is one [12].
The simplex method is a calculation using an iteration path. Before performing iteration calculations, first, change the general form of linear programming to a standard format. Changing the common form begins by changing the equation of the constraint function and adding the initial basis variable to each of the existing constraint functions. The initial basis variable indicates that the activity has not been carried out on the precious resource. The constraint function in general form, even though it is in the form of an equation, must be changed first [14]- [17].
Some things must be considered before changing the general form of simplex into standard form, namely: 1) The inequality with the form ≤ in the general structure of the constraint function is first converted into equation = by adding one slack variable. 2) Change the inequality in general form ≥ into equation = on the constraint function by subtracting one surplus variable.

3) Constraint function with equation = in general form plus artificial variable (artificial variable).
Iterative calculations using the simplex method must be made in tabular form so that the general form that has been converted into standard form is entered into the simplex  It is necessary to pay attention to the steps in solving the simplex method. These steps are [18]- [20]: 1) Changing the objective function with constraints, after all the objective functions are changed, the objective function is changed to an implicit part, 2) Arrange the equations into tabular form.
3) Selecting a key column, the key column to be determined is seen from the objective function line, with the smallest negative value. 4) Selecting the key row, the key row is determined by looking at the smallest ratio value. The ratio value is obtained by dividing the right and key column values. 5) Changing the value of the key row, the value of the key row is changed by dividing all the values in the key row by the key number then there will be an exit variable and an incoming variable. 6) Change the values in the key row with the formula New row = old row -(coefficient per key column * key row value) Continue to improve the steps above until optimal results are found. Optimal results will be obtained when the values in the objective function are all positive.  Figure 1 is a diagram or Flow chart system of 3R furniture where there is some data with different needs or users. The admin can manage all the data. The user only sees the graph, and the cashier can make sales. For Production, 3R Furniture Company faces some constraints. They are raw materials and operational costs. Data processing using linear programming simplex method assisted by WINQSB software shows the results of production optimization obtained by 3R Furniture Company. Based on the production optimization process results, the optimal solution consists of a combination of products, resource status, and sensitivity analysis.

RESULT AND DISCUSSION Production Optimization System Analysis in 3R Furniture
Then, based on the production data provided by the 3R Furniture, there are some products released by the 3R furniture such as Wardrobe, Cupboard, Chairs, Door, Table, windows, etc. Furthermore, those items have different costs, production times, and profits.

Research Design
In data collection, the production capacity data, amount of raw material inventory, the composition of materials for production, and processing time of each product have been obtained. Furthermore, these data will be used to create a mathematical model in the form of a linear program that will be completed with the help of the WINQSB program so that output will be produced that provides optimal information and sensitivity analysis from the existing system discussion.

Decision Variable
A decision variable is a variable that explains the decisions made entirely. Decision Variable will determine the numbers of Wardrobe, Cupboard, Chairs, Door, Table, windows, frames, and tables that must be produced. Based on the Field research, the variable of decision is as follows:

Objectives Function
Objective Function is a function of the decision variable to be maximized or minimized. To assist in making the objective function, it can be seen in table 3.1, namely the product production table. The objective function is as follows = 3900 1 + 450 2 + 200 3 + 400 4 + 50 5 + 100 6 + 700 7

Constraint Function
Constraint Function consists of Raw Materials availability a) Raw Materials Constraints 20 1 + 17 2 + 8 3 + 3 4 + 1.5 5 + 7.5 6 + 1.5 7 ≤ 1755 b) Working Hours Constraints 56 1 + 32 2 + 56 3 + 8 4 + 1 5 + 4 6 + 8 7 ≤ 240 Constraints indicate the capacity of the supply of wood raw materials and working hours for production, raw materials as many as (1755 sheets) and working hours (240 hours). So that the complete formulation model for the problem of the optimum production amount to get the optimal production amount of the product produced at the 3R Furniture Company with the simplex method is as follows:

3) Determining of Pivot Columns
Pivot Columns is a columns that has a row value Z which means as negative for biggest number.

4) Determining Pivot Row
The smallest index is for determining of Pivot row Index= ℎ

5) Determining of new Pivot row element
Current Pivot row = Pivot row / Pivot number Table 8. Determining of new pivot row value  NB  X1  X2  X3  X4  X5  X6  X7  S1 S2 NK Z S1    Figure 3. Iteration Table 1 Figure 4. Iteration table 2 The calculation of profit optimization with the simplex method obtained maximum results, namely profits obtained by using all available production time and labor to produce tables (X7), namely a total of 30 units of tables resulting in a gain of Rp. 20999.998. Based on these results, it can be seen that the linear programming method can be used in profit optimization. These results are related to research that uses the linear programming method to solve optimization cases.
In this study, the validation was carried out by testing. This test is done by involving calculations. This test aims to measure the level of truth of the calculation results of the system made. The test method is by performing calculations with different inputs or case studies, and the accounting results will be matched with data from the case studies. The test table can be seen in  Based on the test results contained in table 10, it can be concluded that as many as 10 data samples produce accuracy values according to the following calculations: Accuracy element counting = 10/10*100= 100%. So based on the results of tests carried out on the system with as much as 10 data, the accuracy value obtained is 100% accurate, which shows that the optimization system is functioning correctly.

CONCLUSION
Based on the findings and results, several conclusions can be drawn that the mathematical model for optimizing cost production in 3R Furniture consists of the objective and constraint functions. The objective function is maximizing the sum of the multiplication between the contributions of production costs with the decision variables of each style on each machine. The constraints function is the constraints of working hours, demand, and use of wood. The working hours constraints for each device must be less than 24 hours, the demand constraints for each style produced on the seven machines are at least the same or more than the order, and the wood capacity constraints indicate that the seven types made in each production must less than or equal to the production capacity for each exhibition, and the decision variable must be non-negative. The working hours constraint for each machine must be less than 24 hours, the demand constraints for each style produced on the seven machines are at least the same or more than the order, and the wood capacity constraints indicate that the seven types made in each production must less than or equal to the production capacity for each show, and the decision variable must be non-negative.
This study also concluded that the cost of producing 3R furniture by the 3R Furniture Company is optimal. It can be shown by the calculation of the program that I made that fulfills the objective function and the constraint function. In other words, the result obtained is the optimal value. In addition, because the profit from the calculation carried out by the 3R Furniture Company is Rp. 20,999,998.00 from the table production carried out by the 3R Furniture Company, it can be concluded that the table production profit by the 3R Furniture Company is optimal.