Math Problem Statement

In the process of using the simplex method to solve a linear programming maximization problem, the table below was obtained. What row operation should be performed to make the -10 into a zero?

Solution

To make the -10 in the ZZ-row a zero using the simplex method, we use the pivot row (the yy-row here). The correct operation is:

"Multiply the pivot row by 10 and then add it to the row with the -10".

Why?

  1. The pivot element in the yy-row for yy is 1 (highlighted).
  2. To eliminate 10-10 in the ZZ-row, multiply the pivot row (yy-row) by 10.
  3. Add the result to the ZZ-row:
    Znew=Zold+10×(y-row)Z_{\text{new}} = Z_{\text{old}} + 10 \times (y \text{-row}).

This operation makes 10-10 into 0.


Would you like a detailed step-by-step calculation for the row operation? Let me know!

Related Questions:

  1. What is the pivot element, and how is it chosen in the simplex method?
  2. Why is the pivot row normalized before elimination?
  3. How do we determine which column to pivot on in the simplex method?
  4. What is the difference between the simplex method for maximization and minimization problems?
  5. How do degeneracy and cycling occur in the simplex method?

Tip:

In the simplex method, row operations aim to eliminate non-zero coefficients in the objective row (top row) while maintaining feasibility in the solution.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Simplex Method
Pivot Row Operations
Row Operations

Formulas

Row operation: New row = Old row + (Multiplication factor × Pivot row)
Pivot row adjustment: Coefficient elimination

Theorems

Simplex Method for Linear Programming

Suitable Grade Level

Undergraduate (Operations Research or Linear Algebra)