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.

Name: To set up the simplex matrix for solving a linear programming problem
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Description: 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.**

Given,Max f = 5 x + 3 y subject to 8 x + 5 y ≤ 300 x + 4 y ≤ 225 and x,y≥0 After introducing slack variablesMax f = 5 x + 3 y + 0 S1 + 0 S2 subject to 8 x + 5 y + S1= 300 x + 4 y + S2 = 225 and x,y,S1,S2≥0Negative minimum Zj-Cj is -5 and its column index is 1. So, the entering variable is x. Minimum ratio is 37.5 and its row index is 1. So, the leaving basis variable is S1. ∴ The pivot element is 8R1→R1÷8R2→R2-R1 Since all Zj-Cj≥0 Hence, optimal solution is arrived with value of variables as : x=37.5,y=0 Max f=187.5

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Set up the simplex matrix used to solve the linear programming problem. Assume all variables are nonnegative. Maximize f = 5x + 3y subject to 8x + 5y < 300 x + 4y s 225. y S1 S2 f first constraint second constraint objective function

Set up the simplex matrix used to solve the linear programming problem. Assume all variables are nonnegative. Maximize f = 5x + 3y subject to 8x + 5y < 300 x + 4y s 225. y S1 S2 f first constraint second constraint objective function

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Topic Video

Question

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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