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Answer :
Final answer:
The optimal solution to the given linear programming problem is x1 = 10, x2 = 5, x3 = 12, with a maximum objective function value of 15.
Explanation:
The given problem is a linear programming problem that can be solved using the Simplex Method. The objective is to maximize the expression x1 - x2 + x3, subject to the following constraints:
- 2x1 + x2 - 3x1 ≤ 40
- x1 + x3 ≤ 25
- 2x2 + 3x3 ≤ 32
- x1, x2, x3 ≥ 0
To solve this problem using the Simplex Method, we need to convert it into standard form by introducing slack variables. The standard form of the problem is:
- Maximize z = x1 - x2 + x3
- Subject to:
- 2x1 + x2 - 3x1 + s1 = 40
- x1 + x3 + s2 = 25
- 2x2 + 3x3 + s3 = 32
- x1, x2, x3, s1, s2, s3 ≥ 0
We can now construct the initial simplex tableau:
Next, we select the most negative coefficient in the bottom row (z-row) as the pivot column. In this case, the most negative coefficient is -1, corresponding to x1. We then select the pivot element, which is the smallest positive ratio of the right-hand side (RHS) to the pivot column coefficient. In this case, the pivot element is 40/2 = 20, corresponding to the entry in the s1-row and x1-column.
We perform row operations to make the pivot column a unit vector and update the tableau:
We repeat the process of selecting the pivot column and pivot element until no further improvements can be made. After several iterations, we obtain the final tableau:
The final tableau shows that the optimal solution is x1 = 10, x2 = 5, x3 = 12, with a maximum objective function value of 15.
Learn more about simplex method here:
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