also can somone explain to me these types of problems.
directions are introduce slacks variables as necessary, then write the intial simplex tableau for each linear programming problem.
find x1 >= 0 and x2 >= 0 such that
x1+x2<=10
5x1+3x2<=75
and z=4x1+2x2 is maximized.
1 2 3 115
2 1 1 200
1 0 1 50
1 1 4 0 is the first augmentation
1 2 1 1
2 1 0 1
3 1 1 4
115 200 50 0 is the second augmentation.
z = -w = -y1-y2-4y3
y1+2y2+3y3 >= 115
2y1 +y2+ y3 <= 200
y1 + y3 >=50
y1 y2 y3 s1 s2 s3 z
1 2 3 1 0 0 0 115
2 1 1 0 -1 0 0 200
1 0 1 0 0 1 0 50
-1 -1 -4 0 0 0 1 0 is the last augmentation and that is as far as I can get
2x1+5x1/10+5x1/100 in decimal number
To solve this linear programming problem, we need to follow a few steps. Let me explain each step to you:
Step 1: Introducing Slack Variables
The first step is to introduce slack variables for each of the inequalities in the problem. Slack variables are non-negative variables that help convert the inequalities into equations. In this case, we have two inequalities.
Inequality 1: x1 + x2 <= 10
Introducing slack variable s1, we rewrite this inequality as an equation: x1 + x2 + s1 = 10
Inequality 2: 5x1 + 3x2 <= 75
Introducing slack variable s2, we rewrite this inequality as an equation: 5x1 + 3x2 + s2 = 75
Step 2: Writing the Initial Simplex Tableau
Now, we can write the initial simplex tableau using the augmented matrix form. The tableau consists of the coefficients of the variables in the equations, the slack variables, and the objective function.
The initial tableau should include the coefficients of x1, x2, s1, s2, and the constant term. Additionally, we need to include the coefficients of the decision variables in the objective function (z = 4x1 + 2x2).
Here's the initial simplex tableau based on the equations we derived:
| x1 | x2 | s1 | s2 | RHS |
------------------------------------
Row 1 | 1 | 1 | 0 | 0 | 10 |
Row 2 | 5 | 3 | 0 | 0 | 75 |
Row 3 | -4 | -2 | 0 | 0 | 0 |
In the initial tableau, the negative coefficients of the objective function reflect that we are trying to maximize the objective function. The RHS column represents the right-hand side constants from the equations.
You can use this initial simplex tableau to perform the simplex algorithm to find the optimal values for x1, x2, and z.
Note: This explanation assumes familiarity with linear programming, the simplex algorithm, and the concept of slack variables. If you are new to linear programming, I recommend studying these topics in more detail to fully understand the process.