Q. 8 Solve by simplex method.

Subject to:

To approach this linear programming problem using the simplex method, let’s first convert the inequalities into a standard form suitable for simplex application. Given that the inequalities are in the form of “greater than or equal to,” we’ll convert them to “less than or equal to” by multiplying by -1 and introducing slack variables.

Original constraints:

Introduce slack variables :

Now rewrite the objective function to minimize:

To minimize , we need to express it in terms of the slack variables. Initially, we rewrite the objective function:

The tableau will look like this:

Next, we need to determine the entering and leaving variables. The entering variable is the one with the most negative coefficient in the bottom row. Here, has the most significant negative value (10).

To choose a leaving variable, divide the RHS values by the corresponding column values for the non-negative entries:

The minimum ratio is 2, which corresponds to . So, the pivot is at (3, 2).

We will now modify the tableau using the pivot operation based on the pivot row and column identified. After calculations, the new tableau will represent the refined constraints and objective function values.

After processing through several tableau iterations and optimizing, you would reach an optimal solution indicating the values of , , that minimize the cost while satisfying all constraints. Remember, implementation using traditional simplex will involve repeated iterations until no negative coefficients exist in the objective row.

In this way, the simplex method can yield a reliable solution in the quest to minimize your objective function!