Math Problem Statement

Find the solutions that can be read from the simplex tableau. The tableau is provided with variables x1, x2, x3, s1, s2, s3, and Z.

Solution

This image shows a simplex tableau, used in linear programming to solve optimization problems. Based on the tableau, I will interpret the basic and non-basic variables and solve for the values of x1,x2,x3,s1,s2,s3x_1, x_2, x_3, s_1, s_2, s_3, and zz.

Simplex Tableau:

& x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & z \\ \hline \text{Row 1} & 7 & 0 & 11 & 6 & 0 & 0 & 44 \\ \text{Row 2} & 0 & 6 & 0 & 4 & 0 & 0 & 42 \\ \text{Row 3} & 0 & 0 & 0 & 2 & 20 & 10 & 10 \\ \text{Objective Row} & -2 & 0 & 0 & 2 & 2 & 4 & 60 \\ \end{array}$$ ### Steps: 1. **Identify basic variables**: Look for columns with a single entry of 1 and the rest being 0s. These variables are basic variables (with non-zero values), while the others are non-basic (set to 0). 2. **Non-basic variables**: $$x_1 = 0$$, $$x_2 = 0$$, $$x_3 = 0$$ (since their columns do not have a leading 1). 3. **Solve for basic variables** using the right-hand side (RHS) values. #### Basic Variable Values: - $$s_1$$ corresponds to Row 1 → $$s_1 = 44$$ (RHS value). - $$s_2$$ corresponds to Row 2 → $$s_2 = 42$$ (RHS value). - $$s_3$$ corresponds to Row 3 → $$s_3 = 10$$ (RHS value). - The value of $$z = 60$$ (from the last row of the tableau). ### Final Solution: $$x_1 = 0, \quad x_2 = 0, \quad x_3 = 0, \quad s_1 = 44, \quad s_2 = 42, \quad s_3 = 10, \quad z = 60$$ Would you like further details on any of the steps or clarifications? Here are some related questions to deepen your understanding: 1. What is the role of slack variables in the simplex method? 2. How do we interpret non-basic variables in linear programming? 3. What does the final value of $$z$$ represent in the context of the simplex method? 4. How can we tell if a solution is optimal from the tableau? 5. What are pivoting steps in the simplex algorithm? **Tip**: Always check if the solution is feasible by ensuring that all slack variables and decision variables meet the problem constraints.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Simplex Method

Formulas

Simplex tableau interpretation
Basic and non-basic variables

Theorems

Fundamental theorem of linear programming
Simplex algorithm

Suitable Grade Level

College level or advanced high school

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