Math Problem Statement
Using simplex method
Max z = x1+2x2
Subject to-x1+2x2<8 x1+2x2<12 x1-x2<3 x1, x2≥0
Solution
To solve the linear programming problem using the Simplex Method, we'll first reformulate the problem by converting the inequalities into equations using slack variables. Let's proceed step by step.
Problem Statement:
Maximize:
Subject to:
-x_1 + 2x_2 & \leq 8 \\ x_1 + 2x_2 & \leq 12 \\ x_1 - x_2 & \leq 3 \\ x_1, x_2 & \geq 0 \end{align*}$$ ### Step 1: Convert inequalities to equalities by introducing slack variables Let $$s_1, s_2, s_3$$ be the slack variables for each inequality: $$\begin{align*} -x_1 + 2x_2 + s_1 &= 8 \\ x_1 + 2x_2 + s_2 &= 12 \\ x_1 - x_2 + s_3 &= 3 \\ x_1, x_2, s_1, s_2, s_3 &\geq 0 \end{align*}$$ ### Step 2: Set up the initial Simplex tableau We express the objective function in standard form: $$Z - x_1 - 2x_2 = 0 \quad \text{(which becomes)} \quad Z + (-x_1 - 2x_2) = 0$$ The initial Simplex tableau looks like this: | Basic Variables | $$x_1$$ | $$x_2$$ | $$s_1$$ | $$s_2$$ | $$s_3$$ | RHS | |-----------------|--------|--------|--------|--------|--------|------| | $$s_1$$ | -1 | 2 | 1 | 0 | 0 | 8 | | $$s_2$$ | 1 | 2 | 0 | 1 | 0 | 12 | | $$s_3$$ | 1 | -1 | 0 | 0 | 1 | 3 | | $$Z$$ | -1 | -2 | 0 | 0 | 0 | 0 | ### Step 3: Perform the Simplex Method 1. **Entering variable**: Identify the most negative coefficient in the objective row. Here, $$x_2 = -2$$ is the most negative, so $$x_2$$ enters the basis. 2. **Leaving variable**: To determine which variable leaves the basis, compute the ratio of the RHS to the corresponding coefficient in the column for $$x_2$$: - Row 1: $$\frac{8}{2} = 4$$ - Row 2: $$\frac{12}{2} = 6$$ - Row 3: $$\frac{3}{-1} =$$ Not considered (negative) So, $$s_1$$ leaves the basis (smallest ratio is 4). 3. **Pivot** on the element at row 1, column $$x_2$$ (which is 2). Let's update the tableau after performing the pivot operation on this element: #### Pivoted Tableau: | Basic Variables | $$x_1$$ | $$x_2$$ | $$s_1$$ | $$s_2$$ | $$s_3$$ | RHS | |-----------------|--------|--------|--------|--------|--------|------| | $$x_2$$ | -0.5 | 1 | 0.5 | 0 | 0 | 4 | | $$s_2$$ | 2 | 0 | -1 | 1 | 0 | 4 | | $$s_3$$ | 0.5 | 0 | 0.5 | 0 | 1 | 7 | | $$Z$$ | -0.5 | 0 | 1 | 0 | 0 | 8 | ### Step 4: Check for optimality There are no negative coefficients in the objective function row (Z-row), so the current solution is optimal. ### StepAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Simplex Method
Optimization
Formulas
Simplex tableau method
Pivot operation in simplex
Slack variables transformation
Theorems
Simplex algorithm for optimization
Suitable Grade Level
College/University level
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