Math Problem Statement

If you would construct the initial simplex tableau, what are the entries in Cj column?

Solution

The image provides a linear programming problem and a multiple-choice question related to the simplex tableau method.

Problem Overview:

The linear program given is:

  • Maximize: P=2x3y+7zP = 2x - 3y + 7z
  • Subject to the following constraints:
    1. 4x+9y104x + 9y \geq 10
    2. x2yz12x - 2y - z \geq 12
    3. 7y+3z87y + 3z \geq 8
    4. 2x92x \geq 9
    5. 3y113y \geq 11
    6. 4z134z \geq 13
    7. x0x \geq 0
    8. y0y \geq 0
    9. z0z \geq 0

Question:

The question asks, "If you would construct the initial simplex tableau, what are the entries in the Cj column?"

Answer Choices:

  1. Coefficients of variables in the modified objective function that appear in the basis column
  2. Coefficients of variables in the modified objective function
  3. Coefficients of variables in the objective function
  4. Coefficients of variables in the objective function that appear in the basis column
  5. Coefficients of variables in the Cj row

Answer Explanation:

In the initial simplex tableau, the Cj column refers to the coefficients of the variables in the objective function P=2x3y+7zP = 2x - 3y + 7z.

The correct choice would be:

Option 3: "Coefficients of variables in the objective function."

This is because the Cj column typically contains the coefficients of the decision variables (in this case, x,y,zx, y, z) from the objective function, which are 2,3,72, -3, 7.

Would you like to explore more details about the simplex method?

5 Relative Questions:

  1. How do you convert inequalities to equalities when constructing the simplex tableau?
  2. What is the role of slack, surplus, and artificial variables in the simplex method?
  3. How do you select the pivot element in the simplex method?
  4. What does the optimality condition in the simplex method imply?
  5. Can the simplex method handle minimization problems directly?

Tip:

When forming the initial simplex tableau, make sure to transform all inequalities into equalities by introducing slack or surplus variables.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Simplex Method
Objective Function
Inequality Constraints

Formulas

Objective Function: P = 2x - 3y + 7z
Inequality Constraints
Simplex Tableau Construction

Theorems

Fundamental Theorem of Linear Programming
Simplex Algorithm

Suitable Grade Level

University Level (Operations Research, Linear Programming)