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 Z=15x+10yZ = 15x + 10y

Subject to: 4x+6y3604x + 6y \leq 360 3x1803x \leq 180 5y2005y \leq 200 x0,y0x \geq 0, \quad y \geq 0

Step 1: Convert inequalities into equalities using slack variables

For each constraint, we introduce a slack variable:

  1. 4x+6y+s1=3604x + 6y + s_1 = 360, where s1s_1 is the slack variable for the first inequality.
  2. 3x+s2=1803x + s_2 = 180, where s2s_2 is the slack variable for the second inequality.
  3. 5y+s3=2005y + s_3 = 200, where s3s_3 is the slack variable for the third inequality.

Thus, the system becomes:

  • 4x+6y+s1=3604x + 6y + s_1 = 360
  • 3x+s2=1803x + s_2 = 180
  • 5y+s3=2005y + s_3 = 200

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 Z=15x+10yZ = 15x + 10y, we need to maximize ZZ, so we represent this as Z15x10y=0Z - 15x - 10y = 0.

Now, set up the tableau:

Basic Variablexxyys1s_1s2s_2s3s_3RHS (Right-Hand Side)
s1s_146100360
s2s_230010180
s3s_305001200
ZZ-15-100000

Step 3: Apply the Simplex method

  1. Identify the pivot column: Choose the most negative value in the objective row (last row). Here, the most negative coefficient is -15 (for xx).

  2. Identify the pivot row: Divide the RHS by the corresponding coefficient in the pivot column. The minimum ratio determines the pivot row:

    • For s1s_1: 3604=90\frac{360}{4} = 90
    • For s2s_2: 1803=60\frac{180}{3} = 60
    • For s3s_3: 2005=40\frac{200}{5} = 40

    The minimum ratio is 40, so the pivot row is s3s_3, and the pivot column is xx.

  3. 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 xx and yy 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