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(d) Maximize $9x_2 + 2x_3 - x_5$ subject to:
$x_1 - 3x_2 - 4x_4 + 2x_6 = 60$
$2x_2 - x_4 - x_5 + 4x_6 = -20$
$x_2 + x_3 + 3x_6 = 10$
$x_1, x_2, x_3, x_4, x_5, x_6 \geq 0$

Answer :

To solve this linear programming problem, we need to maximize the objective function:

Maximize: [tex]9x_2 + 2x_3 - x_5[/tex]

subject to the following constraints:

  1. [tex]x_1 - 3x_2 - 4x_4 + 2x_6 = 60[/tex]

  2. [tex]2x_2 - x_4 - x_5 + 4x_6 = -20[/tex]

  3. [tex]x_2 + x_3 + 3x_6 = 10[/tex]

  4. [tex]x_1, x_2, x_3, x_4, x_5, x_6 \geq 0[/tex]

This is a standard linear programming problem that can be solved using the Simplex method or a software tool like MATLAB, Python (with libraries such as scipy or PuLP), or a dedicated optimization tool like LINDO.

Step-by-step Explanation:

  1. Convert Constraints into Equations:

    • Ensure the equations are in equality form, which they already are.
    • Make sure all variables are non-negative, which is specified here.

  2. Set up the Initial Simplex Tableau:
    The tableau method requires constructing an initial tableau from the equations. This involves ensuring that each equation has a basic variable.

  3. Identify Basic and Non-basic Variables:
    A basic variable solution starts with selecting a set of initial basis variables that can be set to zero. The remaining variables are adjusted in terms of these basis variables.

  4. Iterate to Improve the Objective Function:
    Use the Simplex method to perform pivot operations which adjust the tableau to continually improve the objective function.

  5. Continue Until Optimal Solution is Found:
    When the objective function cannot be improved further, an optimal solution is found, giving values for [tex]x_1, x_2, x_3, x_4, x_5, x_6[/tex].

Example Solution Process:

For those using software:

  • You would input these equations and the objective function into the LP solver.
  • The tool will carry out the iterations and return values for the variables that maximize the objective function.

For those solving manually:

  • You must pivot manually, which is complex and usually covered in detail in linear programming courses.

It's recommended to use software tools for quicker and error-free solutions unless the problem is small or educational methods require a manual solution.

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