Math Problem Statement
(2pts) A simplex matrix is given in which the solution is complete. Identify the maximum value of 𝑓 and a set of values of all the variables that gives this maximum value. 𝑥 𝑦 𝑠1 𝑠2 𝑓 [ 0 4 1 −1/5 0 2 1 2 0 1 0 4 0 2 0 2 1 12
Solution
The given simplex tableau represents a system where the solution is complete. We are tasked with identifying the maximum value of the objective function and the corresponding values of the decision variables.
Here’s the simplex matrix for reference:
0 & 4 & 1 & -1/5 & 0 & 2 \\ 1 & 2 & 0 & 1 & 0 & 4 \\ 0 & 2 & 0 & 2 & 1 & 12 \end{bmatrix}$$ Let’s break this down step by step. ### Step 1: Interpret the Tableau - The tableau contains columns corresponding to variables $$x$$, $$y$$, and slack variables $$s_1$$, $$s_2$$, as well as the value of the objective function $$f$$ in the last column. - The basic variables are determined by looking at columns that resemble the identity matrix (i.e., columns with exactly one "1" and all other entries "0"). - Column 2 corresponds to variable $$x$$ since it has a "1" in the second row and "0"s elsewhere. - Column 4 corresponds to slack variable $$s_2$$. Thus, $$x$$ and $$s_2$$ are basic variables, while $$y$$ and $$s_1$$ are non-basic variables. ### Step 2: Identify the Values of the Variables - For basic variables, their values are given by the right-hand side (RHS) of the tableau. - $$x = 4$$ (from row 2). - $$s_2 = 12$$ (from row 3). - The non-basic variables $$y$$ and $$s_1$$ are zero since they do not correspond to basic variables. Thus, the values of the variables are: - $$x = 4$$ - $$y = 0$$ - $$s_1 = 0$$ - $$s_2 = 12$$ ### Step 3: Determine the Maximum Value of $$f$$ The maximum value of the objective function $$f$$ is found in the first row of the last column, which gives: $$f = 2$$ ### Final Answer: - The maximum value of $$f$$ is **2**. - The corresponding values of the variables that give this maximum value are: - $$x = 4$$ - $$y = 0$$ - $$s_1 = 0$$ - $$s_2 = 12$$ Would you like a more detailed breakdown of any specific part of the solution? Here are some related questions to explore further: 1. What is the significance of the slack variables in the simplex method? 2. How do you determine which variables are basic and non-basic in a simplex tableau? 3. Can a simplex tableau have multiple solutions for the objective function? 4. How would you interpret the results if $$f$$ were minimized instead of maximized? 5. What steps are involved in pivoting within the simplex algorithm? **Tip:** When solving linear programming problems using the simplex method, always check the feasibility of the solution and ensure that all constraints are satisfied at each step.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Simplex Method
Optimization
Formulas
Objective function f = c1 * x1 + c2 * x2
Simplex tableau analysis for basic and non-basic variables
Theorems
Fundamental Theorem of Linear Programming
Suitable Grade Level
Undergraduate level (Linear Algebra/Operations Research)