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## 4.3: Minimization By The Simplex Method

## Learning Objectives

- Identify and set up a linear program in standard minimization form
- Formulate a dual problem in standard maximization form
- Use the simplex method to solve the dual maximization problem
- Identify the optimal solution to the original minimization problem from the optimal simplex tableau.

From the final simplex tableau, we then extract the solution to the original minimization problem.

## Example \(\PageIndex{1}\)

Convert the following minimization problem into its dual.

To achieve our goal, we first express our problem as the following matrix.

\[\begin{array}{cc|c} 1 & 2 & 40 \\ 1 & 1 & 30 \\ \hline 12 & 16 & 0 \end{array} \nonumber \]

\[\begin{array}{cc|c} 1 & 1 & 12 \\ 2 & 1 & 16 \\ \hline 40 & 30 & 0 \end{array} \nonumber \]

The following maximization problem associated with the above matrix is called its dual.

Note that we have chosen the variables as y's, instead of x's, to distinguish the two problems.

## Example \(\PageIndex{2}\)

Solve graphically both the minimization problem and its dual maximization problem.

Our minimization problem is as follows.

We now graph the inequalities:

## The Duality Principle

## Example \(\PageIndex{3}\)

The initial simplex tableau is

The final simplex tableau reads as follows:

We restate the solution as follows:

The minimization problem has a minimum value of 400 at the corner point (20, 10)

We now summarize our discussion.

## Minimization by the Simplex Method

- Set up the problem.
- Write a matrix whose rows represent each constraint with the objective function as its bottom row.
- Write the transpose of this matrix by interchanging the rows and columns.
- Now write the dual problem associated with the transpose.
- Solve the dual problem by the simplex method learned in section 4.1.
- The optimal solution is found in the bottom row of the final matrix in the columns corresponding to the slack variables, and the minimum value of the objective function is the same as the maximum value of the dual.

## Explanation of Simplex Method for Minimization.

Step 2: Determine Slack Variables

Step 3: Setting up the Tableau

Step 5: Identify Pivot Variable

Step 6: Create the New Tableau

Numerical examples are provided below to help explain this concept a little better.

I. To find the s2 value in row 1:

New tableau value = (-3) * (1/5) + 0 = — 3/5

II. To find the x1 variable in row 3:

New value = (10) * (1/5) + -8 = -6

Once the new tableau has been completed, the model can be checked for an optimal solution.

Step 8: Identify New Pivot Variable

With the new pivot variable identified, the new tableau can be created in Step 9.

Step 11: Identify Optimal Values

Non-basic variables: x2, x3, s2

The final solution shows each of the variables having values of:

Let’s put this values in minimization equation :

The maximum optimal value is 64 and found at (8, 0, 0) of the objective function.

Basic variables are variables that are non-negative in terms of the optimal solution.

Non-basic variables are variables that are zero in terms of the optimal solution.

Reference: https://math.libretexts.org/ , Book: Blitzer, Thinking Mathematically | Pearson.

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## Swapnil Bandgar

## Section 4.4 Question 3

How do you apply the simplex method to a standard minimization problem.

The quotients for each row of the tableau are formed below:

Once the pivot is a one, row operations are used to change the rest of the pivot column to zeros.

The entry in the first row of the pivot column is

## Example 3 Find the Optimal Solution

In Section 4.2 , we solved the linear programming problem

using a graph. In Example 2 , we found the associated dual maximization problem,

Apply the Simplex Method to this dual problem to solve the minimization problem.

The initial simplex tableau is formed from the system of equations

To put zeros in the rest of the pivot column, we utilize more row operations.

## IMAGES

## VIDEO

## COMMENTS

The procedure to solve these problems was developed by Dr. John Von Neuman. It involves solving an associated problem called the dual problem.

Step 1: Standard Form · Step 2: Determine Slack Variables · Step 3: Setting up the Tableau · Step 4: Check Optimality · Step 5: Identify Pivot Variable · Step 6:

... https://www.patreon.com/patrickjmt In this video, I show how to use the Simplex Method to find the solution to a minimization problem.

Solving a standard minimization problem using the Simplex Method by create the dual problem. First half of the problem.

Here is the video about LPP using simplex method (Minimization) with three variables, in that we have discussed that how to solve the

Ex 1: Determine a Dual Problem Given a Standard Minimization Problem · Simplex method - Example 5 - Minimization · (New Version Available)

Linear Programming Problem MCQ | LPP MCQ | Operations Research MCQ | Part 1 · Minimization problem Simplex method (Artificial Variable in Simplex

This video shows how to solve a minimization LP problem using the Big M method and the simplex tableau.00:00 Minimization to

The basic procedure used to solve such a problem is to convert it to a maximization problem in standard form, and then apply the simplex method as dis-.

1. Make sure the minimization problem is in standard form. · 2. Find the dual standard maximization problem. · 3. Apply the Simplex Method to solve the dual