Math Problem Statement
Maximize Z = 15x + 10y Subject to 4x + 6y ≤ 360 3x ≤ 180 5y ≤ 200 x ≥ 0, y ≥ 0 solve using simplex method
Solution
To solve the given linear programming problem using the Simplex Method, we follow these steps:
Problem Statement:
Maximize
Subject to:
Step 1: Convert inequalities into equalities using slack variables
For each constraint, we introduce a slack variable:
- , where is the slack variable for the first inequality.
- , where is the slack variable for the second inequality.
- , where is the slack variable for the third inequality.
Thus, the system becomes:
Step 2: Set up the initial simplex tableau
The initial tableau is constructed by arranging the coefficients of the objective function and constraints in a matrix form.
Objective function , we need to maximize , so we represent this as .
Now, set up the tableau:
Basic Variable | RHS (Right-Hand Side) | |||||
---|---|---|---|---|---|---|
4 | 6 | 1 | 0 | 0 | 360 | |
3 | 0 | 0 | 1 | 0 | 180 | |
0 | 5 | 0 | 0 | 1 | 200 | |
-15 | -10 | 0 | 0 | 0 | 0 |
Step 3: Apply the Simplex method
-
Identify the pivot column: Choose the most negative value in the objective row (last row). Here, the most negative coefficient is -15 (for ).
-
Identify the pivot row: Divide the RHS by the corresponding coefficient in the pivot column. The minimum ratio determines the pivot row:
- For :
- For :
- For :
The minimum ratio is 40, so the pivot row is , and the pivot column is .
-
Pivot: Perform Gaussian elimination to make the pivot element (the element at the intersection of the pivot row and pivot column) equal to 1, and update the tableau accordingly.
Step 4: Iterate through the Simplex method
After performing the first pivot operation, the updated tableau will look different, and we will repeat steps 1–3 until all coefficients in the objective function row are non-negative, indicating an optimal solution.
Final Answer
Once the Simplex algorithm converges, the values of and at the optimal solution will be obtained.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Simplex Method
Optimization
Formulas
Maximize Z = 15x + 10y
4x + 6y ≤ 360
3x ≤ 180
5y ≤ 200
x ≥ 0
y ≥ 0
Theorems
Simplex Method
Optimization Theorem
Suitable Grade Level
Grades 11-12
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