The Optimization Problem of Batik Cloth Production with Fuzzy Multi-Objective Linear Programming and Application of Branch and Bound Method

This study discussed fuzzy multi-objective linear programming (FMOLP) and its application. This research was conducted in Rumah Batik Mentari Jambi, which produces five batik motifs typical of the Jambi. In this research, the tolerance for additional raw material capacity is included in the model. This research aims to find out the number of tolerances needed, the maximum number of batik needed to be produced, and the minimum production time so that the producer can earn the maximum profit. The decision variables in FMOLP are the number of pieces of batik measuring in 2m 2 , which means the decision variables must be an integer. Therefore, after obtaining the optimal solution from FMOLP, then proceed with the branch and bound method to obtain the integer solution. The result of this research is that the addition of raw materials needed to earn optimal solutions is as much as 50% of the tolerance assumed in the model. Thus, owner can earn the optimal profit of Rp. 5,675,800.00/week by producing as many as 67 pieces of batik with the design of angso duo, 18 pieces with the design of gentala, and 50 pieces with the design of batang hari, and the minimum production time is 270 hours/week.

is a zeros vector in [18] .

Simplex Method
The steps in the simplex method are as follows [19], [20], [21]: 1) Identify decision variables and formulate them in mathematical symbols; 2) Identify the objective function and constraints; 3) Formulate the objective function and constraints in a mathematical model; 4) Convert the inequality in the constraint into an equation; 5) Input the coefficients on the objective and constraint functions into the simplex table. The coefficients of the objective function are written in the top row; 6) Identify the pivot column, i.e. the most negative entry in the top row; 7) Calculate the quotients which are computed by dividing the value on the right column (the value of ), with the value in pivot column. If denominator is zero or negative, then quotient is ignored. The smallest quotients will be the pivot row; 8) State the pivot element, i.e. the number in intersection of pivot column and pivot row; 9) Make all other entries in pivot column to be zero and perform pivoting. This is just like the method of Gauss-Jordan; 10) The simplex method is finished or stopped if there are no more negative values in the top row.
Otherwise, repeat the algorithm from step 6. 11) When the simplex method is considered finished, then state or identify the basic solution for optimal condition, i.e. look at the columns which have 1 and all other elements zeros. The maximum value appears in the top right hand corner.

Fuzzy Linear Programming (FLP)
The definitions of some of the terms used in Fuzzy Linear Programming (FLP) are as follows [1]: Definition 1: Assume X is a set of objects, then the set containing ordered pairs ̅ = {( , µ ̅ ( )): ∈ } where µ ̅ : → [0,1], is called the fuzzy set in and µ ̅ ( ) is called the membership function.
Definition 2: Assume ̅ is a fuzzy set in and ∈ [0,1] is a real number. Then, a set There are various types of FLP depend on the models and it's solutions that have been summarized by Reza Ghanbari et all in [22]. The formation of the FLP model is derived from the classical LP model in optimization model (1). Let say the value of the objective function ( ) in model (1) is denoted by , and each constraint is modeled as a fuzzy set, then the FLP model for such LP problem (for maximization objective) is generally written as follows e-ISSN: 2686-0341 p-ISSN: 2338-0896 22 Find such that with ∈ , ∈ , ∈ , ∈ × [23]. However, if model (1) has a minimization objective, then the FPL model is generally written as Find such that Notation ≳ is the fuzzy version of ≥ and implies "essentially greater than or equal to". While ≲ is the fuzzy version of ≤ and means "essentially less than or equal to" [24]. Model (2) can be written into a new problem, i.e. to find of the model ≲ ≥ Thus model (3) contains ( + 1) rows [18]. Each inequality in model (3) is represented as a fuzzy set. The membership function of the fuzzy "decision" set of model (3) is ( ) is the membership function of the row -th and can be interpreted as the degree to which satisfies the inequality ≲ , with is row -th of matrix and is row -th of vector [18]. A solution that has the largest membership value is the best solution, so the true optimal solution will be If the boundaries/constraints and objective function are not satisfied, then ( ) = 0. Conversely, if the constraints and objective function are completely fulfilled, then ( ) = 1. One of the simple membership functions that can be used in this situation is for = 1,2, … , + 1, with is a constant that can be chosen subjectively from tolerable violations of the constraints and objective function [18], see Figure 1. If formula (5) is substituted into (4) and with some additional assumptions, we get By introducing a new variable, named , which is basically related to equations (4), (5) and (6), i.e. by defining so that the model becomes [25] maximize with constraint + ≤ + ≥ , By solving the optimization problem in model (7) and assuming that the optimal solution of model (7) is ( , * ), then we can say that * is the optimal solution for the model (2) based on the assumption of membership function (5). Furthermore, − can be found from equation = 1 − , with + is the constant in right-hand side of the th constraint [18].

Fuzzy Multi-objective Linear Programming (FMOLP)
There are many fields of science where optimal decisions depend on two or more objectives, so that they apply multiobjective optimization. An optimization problem which involves more than one objective functions are called a multiobjective optimization problem [26]. Fuzzy multi-objective linear programming (FMOLP) is an optimization method that has more than one objective function subject to multiple constraints. The solution of an FMOLP problem can be obtained using the same method as an LP problem that has one objective function [23]. Other methods which also can be used to solve FMOLP problem are epsilon-constraint method [27], by using fuzzy dominant degrees [28], by using game theory approach [29], and the two-stage method which in the first stage, the FMOLP is transformed into the interval FMOLP, and then transformed again into the crisp multiobjective LP. Then in the second stages, the crisp multi-objective LP is converted into monoobjective program [30], and other methods as explained in [31] and [32] .

Branch and Bound Method
The Branch and Bound method is a well-known technique applied to solve Integer Programming problems, which is a problem that requires the decision variable to be an integer. The basic concept of this method is the branching stage and the stage of determining the boundaries for the existence of feasible solutions (bounding) [13]. Here is the algorithm of Branch and Bound method for maximization problem [14]. 1) Set the initial lower bound, i.e. = −∞ and = 0.

2) Fathomed/bounding stage.
Choose a that is a sub-problem to be tested and found a solution. Next, try to achieve a fathomed condition through one of the following conditions: a) The value of , which is the optimal value of , cannot produce an objective value that is better than the current lower bound value, b) produces an integer solution that is feasible and better than the current lower bound, c) has no feasible solution.
As a result, there will be two possibilities: a) If fathomed and a better feasible solution is found, then the lower bound should be updated. z must be updated. If all sub-problems have reached the fathomed state, then stop branching, the lower bound is said to have provided the optimal solution. If this is not the case, then set = + 1. And repeat step 2). b) If L Pi is not fathomed, then go to step 3) for branching.

3) Branching stage
Choose a variable (whose constraint is integer) whose optimal value is * in the solution of the is non integer, such that ≤ ⟦ * ⟧ and ≥ ⟦ * ⟧ + 1.

Results and Discussion
In this research, there are five decision variables for modelling the optimization problem. They express the number of batik cloths to be produced with 5 different motifs, namely 1 = production number of batik motif tampuk manggis per week 2 = production number of batik motif duren pecah per week 3 = production number of batik motif angso duo per week 4 = production number of batik motif gentala per week 5 = production number of batik motif per week The length of each piece of batik cloth is 2m 2 .
The data used in this research are primary data from Rumah Batik Mentari, which produces five batik motifs. The main raw materials to produce batiks are fabric, wax and fabric dye. The use or consumption of raw materials for each batik motifs is presented in Table 1. Based on the data in Table 1, we want to analyze the optimal solutions if we want to maximize the profit and minimize the production time to avoid overtime. To achieve this goals, then we build a mathematical model in the form of multi-objective linear programming where the first objective is maximizing profit ( 1 ), while the second objective is minimizing production time ( 2 ). For this research, the owner of Rumah Batik Mentari did not state the minimum production for each motives. Therefore, the multi-objective linear programming is as follow: Maximize .
Let means the entry in ℎ -rows in matrix . 3 comes from the coefficient of the decision variables in constraint (a) in model (8), 4 is the coefficient in constraint (b) in model (8) and 5 is the coefficient in constraint (c) in model (8). Furthermore, if a tolerance is added to each constraint, namely an additional maximum of 60 meters for fabric capacity, an additional maximum of 40 ounces for wax capacity, and an additional 400 grams for dye capacity, then model (8)  From model (9), for the sake of consistency of the index for constraints, it can be said that 3 = 60, 4 = 40 and 5 = 400. The state = 0 indicates no addition of raw materials and = 1 indicates the addition of raw materials. The optimal solution of the model (9) in the state = 0 an = 1 can be found by the simplex method, which is shown in Table 2.  (10) is taken from the maximization objective in model (9) and the constrain (b) is taken from the minimization objective in model (9). While the constraints (c), (d) and (e) in model (10) are respectively derived from the constraints (a), (b) and (c) in model (9), respectively. The optimal solution of model (10) is found using the two-phase technique because there is an inequality in the form of greater than equal (≥) in the constraints. The solution of model (10) is shown in Table 3  Table 3. Solution of model (10) Solution Since = 0.5, then we obtain = 1 − 0.5 = 0.5. By substituting = 0.5 into model (9), then the total of inventory capacity required is Fabric capacity: 3 = 2 1 + 2 2 + 2 3 + 2 4 + 2 5 = 270.
The membership function on the constraints is used to determine the degree of membership value for each constraint. Based on the membership function formula in equation (5), for this case, the membership function value for each constraint of model (8) is obtained as follows The membership degree value for each constraint of 0.5 indicates that, the addition of raw materials needed to obtain the optimum solution is 0.5 times the tolerance of each constraint. That is, for fabric capacity an additional 0.5 × 60 = 30 meters is needed. For the wax capacity, it takes an additional 0.5 × 40 = 20 ounces of wax. And for the dye capacity, it takes an additional 0.5 × 400 = 200 grams of dye.
Thus, if the constants in right-hand side in the first constraint, the second constraint and the third constraint respectively are applied 270, 320 and 6,000 to model (9), then using the simplex method, the optimal solution can be obtained, namely 1 = 0, 2 = 0, 3  The decision variables ; = 1,2,3,4,5 in this optimization problem is the number of fabric in pieces (2m 2 ), so an integer solution is required in this problem. Unfortunately, FMOLP gives a noninteger solutions, so it is necessary to apply the Branch and Bound method to the model by adding the integer constraints 1 , 2 , 3 , 4 and 5 . The Branch and Bound chart to find the integer solution is shown in Figure 2 with = 0.5. Through the Branch and Bound method, for the purpose of maximizing 1 and minimizing 2 , the optimum solution is finally obtained in sub-problem 36 because in this sub-problem, the value of 1 is the highest, while the value of 2 is the lowest, with condition that all decision variables ( 1 , 2 , 3 , 4 , 5 ) have integer value, i.e 1 = 0, 2 = 0, 3 = 67, 4 = 18, 5 = 50, 1 = 5,675,800, 2 = 270.

Conclusion
The optimization problem of batik cloth production at Rumah Batik Mentari is analyzed through the Fuzzy Multi Objective Linear Programming (FMOLP) model by considering two objective functions, namely maximizing profit and minimizing production time. The optimum solution of the FMOLP is to produce batik cloth with a total of 135 pieces of batik cloth consisting of 67 pieces of angso duo motif, 18 pieces of gentala motif and 50 pieces of batang hari motif. Meanwhile, the motifs of tampuk mangosteen and durian pecah do not contribute to gain maximum profit based on current data. With this result, maximum profit can be obtained is IDR 5,675,800 / week with the required production time of 270 hours / week. Eventhough current result might be unsatisfactory for the owner because there are two motifs that do not contribute to gain maximum profit based on this research analysis, but due to some people's interest in such two motifs, the owner still produce batik with motifs of tampuk mangosteen and durian pecah in few number of production. While in the meantime, the owner needs to review the capacity comparing to materials consumptions such that the analysis will give result of optimal solution for all motifs.