Math Problem Statement
Solution
Let's go step-by-step to determine the answers.
(A) Entering Variable
The entering variable is determined by selecting the variable in the objective row (bottom row of the tableau) with the most negative coefficient. This indicates the direction to increase .
From the objective row:
Here, is the most negative coefficient, and it corresponds to .
Answer: is the entering variable.
(B) Calculate the Ratios
To determine the exiting variable, we calculate the ratios for each row using:
- Identify the column for (the entering variable). This is the 5th column in the tableau.
- For each row, divide the RHS (last column) by the coefficient in the column, but only if the coefficient is positive. If the coefficient is zero or negative, the ratio is DNE (Does Not Exist).
The tableau:
& x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & Z & \text{RHS} \\ \hline \text{Row 1} & 8 & 6 & 0 & 1 & 3 & 0 & 0 & 174 \\ \text{Row 2} & 0 & 1 & 1 & 0 & 5 & 0 & 0 & 196 \\ \text{Row 3} & 8 & 8 & 0 & 0 & 3 & 1 & 0 & 77 \\ \text{Objective} & 1 & -1 & 0 & 0 & -2 & 0 & 1 & 57 \end{array}$$ Now calculate the ratios: - **Row 1**: Coefficient of $$s_2 = 3$$ (positive). $$\text{Ratio} = \frac{174}{3} = 58.00$$ - **Row 2**: Coefficient of $$s_2 = 5$$ (positive). $$\text{Ratio} = \frac{196}{5} = 39.20$$ - **Row 3**: Coefficient of $$s_2 = 3$$ (positive). $$\text{Ratio} = \frac{77}{3} = 25.67$$ --- | Row | Basic Variable | Ratio as Decimal | |-------|----------------|------------------| | Row 1 | $$s_1$$ | $$58.00$$ | | Row 2 | $$x_2$$ | $$39.20$$ | | Row 3 | $$x_1$$ | $$25.67$$ | --- ### (C) **Exiting Variable** The exiting variable is the one with the **smallest positive ratio** calculated above. This ensures feasibility after the pivot. From the ratios: - Row 1: $$58.00$$ - Row 2: $$39.20$$ - Row 3: $$25.67$$ The smallest positive ratio is $$25.67$$, which corresponds to **Row 3**. The basic variable in Row 3 is $$x_1$$. **Answer:** $$x_1$$ is the exiting variable. --- ### Final Answers: 1. **Entering variable:** $$s_2$$ 2. **Ratios:** - Row 1: $$58.00$$ - Row 2: $$39.20$$ - Row 3: $$25.67$$ 3. **Exiting variable:** $$x_1$$ --- ### Related Questions: 1. What happens to the tableau after performing the pivot operation? 2. How do you determine optimality in the simplex method? 3. What is the significance of selecting the entering variable with the most negative coefficient? 4. What if no positive ratios exist for the exiting variable? 5. Can degeneracy occur in this tableau? Why or why not? **Tip:** Always verify calculations for the entering and exiting variables carefully, as mistakes here can affect the entire optimization process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Simplex Method
Pivot Operations
Optimization
Formulas
Ratio = RHS / Coefficient of entering variable (when coefficient > 0)
Select entering variable: Most negative coefficient in the objective row
Exiting variable: Smallest positive ratio
Theorems
Simplex Method Optimality Condition
Feasibility Rule for Pivoting
Suitable Grade Level
Undergraduate (Optimization Methods, Operations Research)