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X1 X2 X3 X4 rhs 1 2 1 -1 1 2 1 -2 3 3

X1 X2 X3 X4 rhs 1 2 1 -1 1 2 1 -2 3 3

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider an LP with the optimal tableau shown in table below.

a)Does this LP have more than one BFS that is optimal?b)How many optimal solutions does this LP have? ( Hint: if the value of x3 increased, then how does this change the values of the basic variables and z-value?)

 

This table is part of a system of linear equations commonly used in operations research and linear programming, represented as a tableau. The rows represent different constraints, while the columns correspond to variables and the right-hand side (rhs) of the equations. ### Tableau | | \( z \) | \( x_1 \) | \( x_2 \) | \( x_3 \) | \( x_4 \) | rhs | |------|---------|-----------|-----------|-----------|-----------|-----| | Row 1| 1 | 0 | 0 | 0 | 2 | 2 | | Row 2| 0 | 1 | 0 | -1 | 1 | 2 | | Row 3| 0 | 0 | 1 | -2 | 3 | 3 | **Explanation:** - **\( z \):** Typically represents the objective function value in a standard linear programming problem. - **\( x_1, x_2, x_3, x_4 \):** Decision variables in the problem. - **rhs:** Stands for "right-hand side," which indicates the constant part of the equation on the right of the equality or inequality. **Interpretation:** - The first row might represent the objective function in a maximization problem where the coefficients signify contributions per unit of decision variables. - Subsequent rows (Rows 2 and 3) display constraints of the system, in which each decision variable has a specific impact, adjusted by coefficients.
Transcribed Image Text:This table is part of a system of linear equations commonly used in operations research and linear programming, represented as a tableau. The rows represent different constraints, while the columns correspond to variables and the right-hand side (rhs) of the equations. ### Tableau | | \( z \) | \( x_1 \) | \( x_2 \) | \( x_3 \) | \( x_4 \) | rhs | |------|---------|-----------|-----------|-----------|-----------|-----| | Row 1| 1 | 0 | 0 | 0 | 2 | 2 | | Row 2| 0 | 1 | 0 | -1 | 1 | 2 | | Row 3| 0 | 0 | 1 | -2 | 3 | 3 | **Explanation:** - **\( z \):** Typically represents the objective function value in a standard linear programming problem. - **\( x_1, x_2, x_3, x_4 \):** Decision variables in the problem. - **rhs:** Stands for "right-hand side," which indicates the constant part of the equation on the right of the equality or inequality. **Interpretation:** - The first row might represent the objective function in a maximization problem where the coefficients signify contributions per unit of decision variables. - Subsequent rows (Rows 2 and 3) display constraints of the system, in which each decision variable has a specific impact, adjusted by coefficients.
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