Sample Problem and Solving Mathematical Model Of A Linear Program
Mathematical Model - In the previous post we both learn about Understanding Linear Programs And Mathematics Model SMA 11th Class . Therefore, The basic mathematical formula will continue with the material this time by presenting several examples of questions about the mathematical model. The mathematical model is a mathematical formula derived from a process of interpreting an everyday event into a mathematical formula or language. In order to better understand how to construct a mathematical model of a linear programming problem, consider the following examples:
Sample Problem and Solving Mathematical Model Of A Linear Program
Sample Problem 1:
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A factory produces two types of goods K and L using two machines namely G1 and G2. The goods K, G1 engine must operate for 3 minutes and G2 engine for 6 minutes. As for producing L goods, the G1 engine must operate for 9 minutes and the G2 engine runs for 6 minutes. G1 and G2 engines can only operate no more than 9 hours a day. The net profit earned for each item K is Rp.350 and for each item L is Rp.700.
Try to make a mathematical model of the linear program problem, if the expected net gain is maximized.
[1945907] [194590]]
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The description on the above problem can be written in the table as follows:
| | Items K | Goods L | Operation every day |
| Machine G1 | 3 Minutes | 9 Minute 540 Minutes | |
| The G2 engine | 6 Minutes | 6 Minutes | 540 Minutes |
| Advantages | Rp. 350 | Rp. 700 |
We Suppose that the goods of K are produced p the fruits and goods of L are produced q fruit, then:
Operating time required for machine G2 = 6p + 6q
Since the G1 and G2 engines must not operate for more than 9 hours = 540 minutes per day, they must be met in the following inequality:
] 3p + 9q ≤ 540 -> p + 4q ≤ 180
6p + 6q 540 -> p + q
Keep in mind that p and q represent the number of goods, then p and q Can not be negative and its value must be a count. Thus, p and q must satisfy the inequalities below:
p 0, q ≥ 0, and p and q ε C
The net profit earned in Rupiah = 350p + 700q and the expected net profit is maximized. So the mathematical model that can be formed based on the above problem is:
p ≥ 0, q ≥ 0, p + 4q ≤ 180, and p + q ≤ 90; P and q ε C
With the form ( 350p + 700q ) as much as possible.
Sample Problem 2:
A pharmaceutical manufacturer provides two types of mixtures of L and M. The basic ingredients contained in each Kilogram of mixed L and M can be seen in the following table:
| | | |
| Material 2 | ||
| Mixed L | 0.4 Kg | 0.6 Kg |
| | 0.8 Kg | |
From a mixture of L and M there will be a mixture of N. Mixtures N contains at least 1 Kg of material and 3Kg of material 2. The price of each kilogram of L mixture is Rp. 30,000 and each M mixture is Rp. 15,000.
Determine the mathematical model of the above equation if the total cost of making the N mixture is expected to be as cheap as
Solution:
Suppose the N mixture is made of x Kg mixture of L and y Kg M mixture,
Material 1 contained = 0.4x + 0.8y
Since at least 1 ingredient contains 1 Kg, It must be satisfied by the following inequalities:
0.4x + 0.8y ≥ 4 Kg -> x + 2y ≥ 10
Material 2 contained = 0.6x + 0.2y
Since it contains at least 3 Kg of material, it must be satisfied by the following inequality:
0.6x + 0, 2y ≥ 3 Kg -> 3x + y ≥ 15
It is known that x and y represent the weight of the mixture so that its value can not be negative and must be expressed in terms of real numbers. Therefore, x and y are required to satisfy the same inequalities below:
x ≥ 0, y ≥ 0, x And y ε R
The total cost required to make the mixture N = 30000x + 15000y with total cost expected to be as cheap as cheap. Then the mathematical model is: [1945909]
x ≥ 0, y ≥ 0, x + 2y ≥ 10, and 3x + y ≥ 15; X and y ε R
With the form ( 30000x + 15000y ) as small as possible
[1945909]
That's 2 pieces Example Problem and Solving Mathematical Model Of A Linear Program hope it helps you To be better able to understand high school mathematics subject matter on mathematical model and also can make you more understanding about the procedures and steps that must be done to solve similar problems. The spirit continues to learn mathematics !!!
