Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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To set up the simplex matrix for solving a linear programming problem, consider the following: **Objective:** Maximize \( f = 5x + 3y \) **Subject to Constraints:** 1. \( 8x + 5y \leq 300 \) 2. \( x + 4y \leq 225 \) Assume all variables are nonnegative. **Simplex Matrix Structure:** The simplex matrix is organized in tabular format with columns representing variables (\( x, y, s_1, s_2, f \)) and rows representing the constraints and objective function: - The top row is labeled \( x, y, s_1, s_2, f \) for each variable including the slack variables \( s_1 \) and \( s_2 \). - The next rows are organized as follows: - **First Constraint:** Corresponds to \( 8x + 5y + s_1 = 300 \) - **Second Constraint:** Corresponds to \( x + 4y + s_2 = 225 \) - **Objective Function:** Corresponds to maximizing \( f = 5x + 3y \) Each cell in the matrix will eventually be filled with coefficients from the constraints and objective function. The matrix includes a vertical line separating the coefficient matrix from the right-hand side (RHS) values, which contain the constants from each constraint and the objective function value to be maximized. **Explanation for Graphical Representation:** The graphical representation is a simple rectangular matrix with empty cells beside each label and beneath the \( x, y, s_1, s_2, f \) columns. Each label describes its respective role within the simplex matrix, where slack variables (\( s_1, s_2 \)) are added to convert inequalities into equalities. The matrix is used to carry out the simplex method iteratively until an optimal solution is reached. This setup allows for systematic row operations to find the values of \( x \) and \( y \) that maximize the objective function \( f \) under the given constraints.
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Transcribed Image Text:To set up the simplex matrix for solving a linear programming problem, consider the following: **Objective:** Maximize \( f = 5x + 3y \) **Subject to Constraints:** 1. \( 8x + 5y \leq 300 \) 2. \( x + 4y \leq 225 \) Assume all variables are nonnegative. **Simplex Matrix Structure:** The simplex matrix is organized in tabular format with columns representing variables (\( x, y, s_1, s_2, f \)) and rows representing the constraints and objective function: - The top row is labeled \( x, y, s_1, s_2, f \) for each variable including the slack variables \( s_1 \) and \( s_2 \). - The next rows are organized as follows: - **First Constraint:** Corresponds to \( 8x + 5y + s_1 = 300 \) - **Second Constraint:** Corresponds to \( x + 4y + s_2 = 225 \) - **Objective Function:** Corresponds to maximizing \( f = 5x + 3y \) Each cell in the matrix will eventually be filled with coefficients from the constraints and objective function. The matrix includes a vertical line separating the coefficient matrix from the right-hand side (RHS) values, which contain the constants from each constraint and the objective function value to be maximized. **Explanation for Graphical Representation:** The graphical representation is a simple rectangular matrix with empty cells beside each label and beneath the \( x, y, s_1, s_2, f \) columns. Each label describes its respective role within the simplex matrix, where slack variables (\( s_1, s_2 \)) are added to convert inequalities into equalities. The matrix is used to carry out the simplex method iteratively until an optimal solution is reached. This setup allows for systematic row operations to find the values of \( x \) and \( y \) that maximize the objective function \( f \) under the given constraints.
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