Math Problem Statement
Solve the linear programming problem below using the simplex algorithm. Include all the solution steps. min π = 5π₯ β 4π¦ subject to π₯ + π¦ β€ 5 2π₯ β π¦ = 16 π₯ β₯ 0 π¦ β β
Solution
To solve the given linear programming problem using the simplex algorithm, we'll follow the steps systematically. Here is the problem statement:
Objective function:
Subject to the constraints:
Since is not required to be non-negative, we'll have to use a modified simplex approach and carefully handle the equality and inequality constraints.
Step 1: Convert constraints into standard form
1. Convert the inequality into an equation:
For the constraint , introduce a slack variable :
2. Write the equality constraint directly:
So, the problem now looks like:
Objective function:
Subject to:
x + y + s_1 = 5 \tag{1}
2x - y = 16 \tag{2}
Step 2: Set up the initial simplex tableau
We introduce the following variables:
- : Decision variable
- : Decision variable
- : Slack variable
- : Objective function value
To convert the minimization problem into a maximization problem, we rewrite :
The initial tableau is:
Basic | RHS | |||
---|---|---|---|---|
1 | 1 | 1 | 5 | |
2 | -1 | 0 | 16 | |
-5 | 4 | 0 | 0 |
Step 3: Perform simplex iterations
-
Determine the entering variable:
The most negative value in the row is , corresponding to . So, will enter the basis. -
Determine the leaving variable:
Perform the ratio test to find the leaving variable:- For row 1:
- For row 2:
Since 5 is the smallest ratio, leaves the basis.
-
Pivot to form a new tableau:
Pivot on the element in the column of row 1, which is 1.
Basic | RHS | |||
---|---|---|---|---|
1 | 1 | 1 | 5 | |
0 | -3 | -2 | 6 | |
0 | 9 | 5 | 25 |
Step 4: Check for optimality
Since all the coefficients in the row are now non-negative, the current solution is optimal.
Step 5: Extract the solution
From the final tableau:
- Basic variables:
- Slack variable:
Substitute into the original objective function:
Final Answer:
The solution to the linear programming problem is:
- Minimum value of is 1.
Would you like a detailed explanation of any part? Here are some related questions you might find interesting:
- How do we determine the entering and leaving variables in the simplex method?
- What are slack variables and why are they used in linear programming?
- How do we convert a minimization problem into a maximization problem for the simplex method?
- What is the significance of the pivot element in the simplex tableau?
- Can we solve linear programming problems graphically and how does it compare with the simplex method?
Tip: Always double-check your final tableau for non-negative coefficients in the objective function row to confirm optimality!
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Simplex Algorithm
Optimization
Formulas
Objective function: Z = 5x - 4y
Slack variable: x + y + s_1 = 5
Simplex Tableau Method
Theorems
Simplex Method
Linear Programming Standard Form
Suitable Grade Level
Undergraduate
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