Gethuk Manthuk-Manthuk Company makes 2 types of gethuk, namely ENAK and MANTEP. In each production, a maximum of 6 kg of cassava is provided, a minimum of 3 kg of sugar, and 2 kg of coconut.
Example of Solving Problems Using the Graphical Method
To make DELICIOUS gethuk, you need 3kg of cassava, 1kg of sugar, and 2kg of coconut. To make MANTEP gethuk, you need 3kg of cassava, 3kg of sugar, and 1kg of coconut.
Gethuk DELICIOUS is sold for Rp. 5,000,- and Gethuk MANTEP is sold for Rp. 6,000,-
How much gethuk ENAK and gethuk MANTEP should be made? to obtain the MINIMUM production cost, if it is known that the profit for each is Rp. 1000,-
Solution:
1. Determine the variables
| Gethuk | Variabel |
|--------|----------|
| Enak | X1 |
| Mantep | X2 |2. Determine the Objective Function (Z)
Z = … X1 + … X2
| Gethuk | Jual | Keuntungan | Biaya |
|--------|-----------|------------|-----------|
| Enak | Rp.5000,- | Rp.1000,- | Rp.4000,- |
| Mantep | Rp.6000,- | Rp.1000,- | Rp.5000,- |2.b. Simplifying with a comparative scale;
Z = 4000 X1 + 5000 X2
Z = 4X1 + 5X2 → (Using a scale of 1:1000).
3. Determining the Limit Function
| Bahan | Fungsi | Batasan |
|--------|-----------|---------|
| Ketela | 3X1 + 3X2 | ≤ 6 |
| Gula | X1 + 3X2 | ≥ 3 |
| Kelapa | 2X1 + X2 | = 2 |Just a note:
If there is a boundary function in the question, for example x1, x2 ≥ 0, then the graph will have a boundary zone more or less like this;
But if there isn't, then it is infinite, so it is possible that the result will be negative (-).
4. Minimize Z
Three limitation functions have been obtained, including;
- 3X1 + 3X2 ≤ 6
- X1 + 3X2 ≥ 3
- 2X1 + X2 = 2
1). 3X1 + 3X2 ≤ 6
If, X1 = 0, then X2 = 6/3 = 2
If, X2 = 0, then X1 = 6/3 = 2
So, (X1, X2) = (2, 2)
2). X1 + 3X2 ≥ 3
If, X1 = 0, then X2 = 3/3 = 1
If, X2 = 0, then X1 = 3/1 = 3
So, (X1, X2) = (3, 1)
3). 2X1 + X2 = 2
If, X1 = 0, then X2 = 2/1 = 2
If, X2 = 0, then X1 = 2/2 = 1
So, (X1, X2) = (1, 2)
5. Drawing Graphs and Declaring Feasible Regions
Operations Research Graphical Method - Determining Alternative Points
6. Finding the Optimal Z Value
Because the case study asks about the MINIMUM production costs, we only need to use the MINIMUM value, even though at this stage it will also produce the MAXIMUM value.
To find the optimal Z value, 2 methods can be used:
1. By comparing the Z value at each alternative point
The method is to calculate the Z value at each alternative point, by substitution.
Point A
(X1,X2) = (0, 2) substitute into the Objective Function (Z), then;
Z = 4X1 + 5X2
Z = 4*0 + 5*2
Z = 10 (max)Point B, using the 2nd method,
2. By describing the objective function. (trial and error method)
The method is to assume the value of Z in the objective function equation, so that the values of X1 and X2 will be obtained.
Point B
Since point B is the intersection of limits 2 and 3, it is necessary to use both equations/functions,
Elimination:
X1 + 3X2 ≥ 3 | kalikan 2
2X1 + X2 = 2 | kalikan 1
------------------------------
2X1 + 6X2 ≥ 6
2X1 + X2 = 2
------------------------------ (-)
0 + 5X2 = 4
X2 = 4/5Substitute into one of the equations:
X1 + 3X2 ≥ 3
X1 + 3*4/5 ≥ 3
X1 + 12/5 ≥ 3
X1 ≥ 3-12/5
X1 ≥ 15/5 - 12/5
X1 ≥ 3/5So that point B (3/5, 4/5) is obtained. Next, substitute it into the objective function (Z), then;
Z = 4*3/5 + 5*4/5
Z = 12/5 + 20/5
Z = 32/5
Z = 6.4 (min)7. Making Conclusions
So it can be concluded that, to make gethuk ENAKS 3/5 and gethuk MANTEP 4/5, a MINIMUM cost of 6.4 x Rp.1000,- is required (because the scale is 1:1000) = Rp.6400,-
Example 2
Prima bakery company makes 2 types of sandwiches, Legit and Legit Sekali, where in each production, 6 kg of eggs are provided, a maximum of 6 kg of sugar, and a minimum of 2 kg of flour.
To make a Legit sandwich, you need 2kg of Eggs, 3kg of Sugar & 2kg of Flour. While to make a Legit Sekali sandwich, you need 3kg of Eggs, 2kg of Sugar, and 2kg of Flour.
Determine how many Legit and Legit Sekali sandwiches must be made to obtain the cheapest manufacturing cost (MINIMUM), if it is known that Legit sandwiches are sold for Rp. 4,000 and Legit Sekali sandwiches for Rp. 6,000 with a profit of Rp. 1,000 each.
Completion
1. Determine the variables
| Roti Lapis | Variabel |
|--------------|----------|
| Legit | X1 |
| Legit Sekali | X2 |2. Determine the Objective Function (Z)
Z = … X1 + … X2
| Gethuk | Jual | Keuntungan | Biaya |
|--------------|-----------|------------|-----------|
| Legit | Rp.4000,- | Rp.1000,- | Rp.3000,- |
| Legit Sekali | Rp.6000,- | Rp.1000,- | Rp.5000,- |2.b. Simplifying with a comparative scale;
Z = 3000 X1 + 5000 X2
Z = 3X1 + 5X2 → (Using a scale of 1:1000).
3. Determining the Limit Function
| Bahan | Fungsi | Batasan |
|--------|-----------|---------|
| Telur | 2X1 + 3X2 | = 6 |
| Gula | 3X1 + 3X2 | ≤ 3 |
| Terigu | 2X1 + 2X2 | ≥ 2 |4. Minimize Z
Three limitation functions have been obtained, including;
- 2X1 + 3X2 = 6
- 3X1 + 3X2 ≤ 3
- 2X1 + 2X2 ≥ 2
1). 2X1 + 3X2 = 6
If, X1 = 0, then X2 = 6/3 = 2
If, X2 = 0, then X1 = 6/2 = 3
So, (X1, X2) = (3, 2)
2). 3X1 + 3X2 ≤ 3
If, X1 = 0, then X2 = 6/2 = 3
If, X2 = 0, then X1 = 6/3 = 2
So, (X1, X2) = (2, 3)
3). 2X1 + 2X2 ≥ 2
If, X1 = 0, then X2 = 2/2 = 1
If, X2 = 0, then X1 = 2/2 = 1
So, (X1,
5. Drawing Graphs and Declaring Feasible Regions
Determining alternative points of graphical method - example 2 operations research
6. Finding the Optimal Z Value
Because the case study asks about the MINIMUM production costs, we only need to use the MINIMUM value, even though at this stage it will also produce the MAXIMUM value.
To find the optimal Z value, 2 methods can be used:
1. By comparing the Z value at each alternative point
The method is to calculate the Z value at each alternative point, by substitution.
Point A
(X1,X2) = (0, 2) substitute into the Objective Function (Z), then;
Z = 3X1 + 5X2
Z = 3*0 + 5*2
Z = 10 (max)Point B, using the 2nd method,
2. By describing the objective function. (trial and error method )
The method is to assume the value of Z in the objective function equation, so that the values of X1 and X2 will be obtained.
Point B
Since point B is the intersection of limits 1 and 2, it is necessary to use both equations/functions,
Elimination:
2X1 + 3X2 = 6 | kalikan 3
3X1 + 2X2 ≤ 6 | kalikan 2
------------------------------
6X1 + 9X2 = 18
6X1 + 4X2 ≤ 12
------------------------------ (-)
0 + 5X2 = 6
X2 = 6/5Substitute into one of the equations:
2X1 + 3X2 = 6
2X1 + 3*6/5 = 6
2X1 + 18/5 = 6
2X1 = 6-12/5
2X1 = 30/5 - 12/5
2X1 = 12/5
X1 = 12/5 /2
X1 = 12/10
X1 = 6/5So that point B (6/5, 6/5) is obtained. Next, substitute it into the objective function (Z), then;
Z = 3X1 + 5X2
Z = 3*6/5 + 5*6/5
Z = 18/5 + 30/5
Z = 48/5
Z = 9.6 (min)7. Making Conclusions
So it can be concluded that, to make gethuk ENAK 6/5 and gethuk MANTEP 6/5, the MINIMUM cost required is 9.6 x Rp.1000,- (because the scale is 1:1000) = Rp.9600,-
Reference:
CHAPTER II Module Graphic Method (Operations Research Technique), Pengapu Minarwati, ST