The problem in the image presents a linear programming (LP) problem with the objective function:

$\text{Minimize } z = 10x_1 + 8x_2$

subject to the constraints:

$x_1 + 2x_2 \geq b_1, \quad 2x_1 - x_2 \geq b_2, \quad x_1 + 3x_2 \geq b_3$

where $b_1, b_2, b_3$ are constants, and the optimal solution for specific values of these constants is given in the tableau.

The problem asks to determine:

1. The values of $b_1$, $b_2$, and $b_3$ that yield the given optimal solution.
2. The optimal dual solution.
3. The values of constants $a$, $b$, and $c$ in the optimal tableau.

To approach the problem:

• Step 1: Find the values of $b_1, b_2, b_3$: The given tableau shows the final values of the basic and non-basic variables in the optimal solution. From the "Solution" column, we can infer the values of the right-hand side constraints.

• Step 2: Dual solution: Using the primal problem's final tableau, the dual solution can be extracted based on the shadow prices associated with the constraints.

• Step 3: Values of $a, b, c$: These are coefficients within the final tableau, which can be deduced by interpreting the tableau structure.

Next Steps:

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Related Questions:

1. What is the significance of dual variables in linear programming?
2. How are the values of $b_1, b_2, b_3$ related to the constraints in a linear program?
3. Can you explain the process of constructing the simplex tableau for an LP problem?
4. How do shadow prices reflect changes in the objective function?
5. What is the connection between primal and dual solutions in optimization?

Tip:

In linear programming, dual variables represent the rate of change of the objective function with respect to changes in the constraint bounds.