State the dual problem for each linear programming problem....

State the dual problem for each linear programming problem.
Maximize $$z=2 x_{1}+7 x_{2}+4 x_{3}$$
subject to: $$\begin{aligned} 4 x_{1}+2 x_{2}+x_{3} & \leq 26 \\ x_{1}+7 x_{2}+8 x_{3} & \leq 33 \end{aligned}$$
with $$x_{1} \geq 0, \quad x_{2} \geq 0, \quad x_{3} \geq 0$$

Key Concepts

Linear Programming

Linear programming is a method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. It involves optimizing a linear objective function, subject to a set of linear equality or inequality constraints, and is widely used in fields such as operations research, economics, and engineering.

Duality in Linear Programming

Duality is a fundamental concept in linear programming which states that every linear programming problem (called the primal) can be associated with another linear programming problem (called the dual). The solutions to these problems provide bounds for each other, and under certain conditions, the optimal values of both the primal and dual problems are equal.

Primal-Dual Relationship

The relationship between the primal and dual problems is characterized by a one-to-one correspondence between constraints in the primal and variables in the dual, and between variables in the primal and constraints in the dual. This correspondence allows the construction of the dual problem from the primal and ensures that insights from one can be used to understand the other.

Standard Form and Dual Formation

In the context of duality, the standard form of a linear programming problem is crucial. For example, a primal maximization problem with less-than or equal-to constraints and non-negative variables leads to a dual minimization problem with greater-than or equal-to constraints. The process of forming the dual involves transposing the coefficient matrix, switching the roles of the right-hand side values and the objective function coefficients, and reversing the inequality directions.

Transcript

00:01 For this problem, we are being asked to find the dual problem for the maximization equation given here.

00:08 Z equals 2x sub 1 plus 7x up 2 plus 4x of 3.

00:13 And again, we have some constraints given here.

00:16 Now, what does it mean to find the dual problem? well, duality says there is an associated minimization problem that corresponds with a standard maximization problem and vice versa.

00:31 It's an interesting pairing that a minimization and a maximization function will have the same answer.

00:38 So we need to find what the dual problem is.

00:41 What is that related minimization problem that goes with this maximization problem? well, the first thing we're going to do is we're going to put together an augmented matrix that's going to have all of the coefficients of these xs plus we're going to look at our maximization equation itself.

00:58 We're not going to worry about slack variables or anything that we normally do.

01:01 Do with the simplex method, at least not yet.

01:04 So first, what do we have? well, let's start with our conditions.

01:11 And i'm going to take our coefficients, 4, 2, 1, and the constant, which is 26.

01:19 I'm going to do that for both of these.

01:21 The second one is 1, 7, 8, and 33.

01:26 And my last line is going to have the coefficients for my maximization equation itself.

01:33 I'm not going to worry about making them negative like we did in the simplex method.

01:37 I'm just looking at the coefficients themselves.

01:40 And that is 2, 7, 4, and 0.

01:44 Okay.

01:45 Now, we're going to transpose this matrix.