High School

Max z = 90x_1 + 125x_2 + 45x_3 + 65x_4

Subject to:
1. 3x_1 + 3x_2 + x_3 + x_4 <= 72
2. 36x_1 + 48x_2 + 25x_3 + 35x_4 <= 1200
3. x_1 + x_2 <= 500
4. x_3 + x_4 <= 500
5. x_1, x_2, x_3, x_4 >= 0

Solve this problem using the simplex method.

Answer :

The problem you've presented is about solving a linear programming problem using the simplex method. Linear programming is a method to achieve the best outcome, such as maximum profit or lowest cost, within a mathematical model defined by linear relationships. Here, we are maximizing a function given certain constraints.

Let's break it down step-by-step:

  1. Objective Function:
    [tex]\text{Maximize } z = 90x_1 + 125x_2 + 45x_3 + 65x_4[/tex]
    This is the function we want to maximize.

  2. Constraints:

    1. [tex]3x_1 + 3x_2 + x_3 + x_4 \leq 72[/tex]
    2. [tex]36x_1 + 48x_2 + 25x_3 + 35x_4 \leq 1200[/tex]
    3. [tex]x_1 + x_2 \leq 500[/tex]
    4. [tex]x_3 + x_4 \leq 500[/tex]
    5. [tex]x_1, x_2, x_3, x_4 \geq 0[/tex]

  3. The Simplex Method:

    • The simplex method involves converting the constraints into equalities by introducing slack variables. For the constraints above, we introduce slack variables [tex]s_1, s_2, s_3, s_4[/tex].
    • Rewrite the constraints with slack variables:
      1. [tex]3x_1 + 3x_2 + x_3 + x_4 + s_1 = 72[/tex]
      2. [tex]36x_1 + 48x_2 + 25x_3 + 35x_4 + s_2 = 1200[/tex]
      3. [tex]x_1 + x_2 + s_3 = 500[/tex]
      4. [tex]x_3 + x_4 + s_4 = 500[/tex]

  4. Setting up the Initial Simplex Tableau:

    • Set the initial tableau from the above equalities.

  5. Iterative Process:

    • Select the entering variable (using the largest coefficient in the objective function row) and the leaving variable (using the minimum ratio test).
    • Pivot to update the tableau.
    • Repeat the process until there are no more positive coefficients in the objective function row, which indicates an optimal solution.

  6. Interpreting the Solution:

    • The solution provided by the tableau will give values for [tex]x_1, x_2, x_3,[/tex] and [tex]x_4[/tex] that maximize [tex]z[/tex].
    • Ensure non-negativity conditions are met.

The above explanation provides a broad overview of the steps commonly involved in applying the simplex method to solve linear programming problems. For an exact solution, each of these steps would be detailed into rows and columns as in a tableau during an actual calculation session, often performed with computational tools or software for complex problems.

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