Duality in Linear Programming

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Questions and Answers

What is the primary purpose of deriving the dual from the primal in linear programming?

To provide an alternative perspective to analyze the problem (correct)
To simplify the primal problem
To eliminate the need for constraints
To always yield a better solution

When converting a greater than or equal to (≥) type inequality in a maximization problem, which of the following steps should be taken?

Convert to a less than or equal to (≤) type by multiplying by -1 (correct)
Change to an equality and introduce new variables
No changes are required
Directly rewrite it as a less than ( < ) type

What should be done to an equal to (=) type constraint for a minimization problem?

It should be converted directly to a greater than or equal to (≥) type
It can be rephrased as two inequalities with different signs (correct)
It can be rewritten using a single decision variable
It must always be ignored

In the context of linear programming, how should an unrestricted-in-sign decision variable be treated?

<p>It can be rewritten as a difference of two non-negative variables (C)</p> Signup and view all the answers

Which statement is true regarding the conversion of inequalities for a minimization problem?

<p>Less than or equal to (≤) inequalities require no changes (D)</p> Signup and view all the answers

Flashcards

Primal problem

The original form of a linear programming model.

Dual problem

An alternative form of a linear programming model, derived from the primal.

Converting ≥ to ≤

Multiply the greater-than-or-equal-to inequality by -1 to change it to a less-than-or-equal-to inequality.

Converting ≤ to ≥

Multiply the less-than-or-equal-to inequality by -1 to change it to a greater-than-or-equal-to inequality.

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Unrestricted-in-sign variable

A decision variable that can take on both positive and negative values.

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Study Notes

Duality in Linear Programming

Linear programming models have two forms: primal and dual.
The primal is the original form of the model.
The dual is an alternative model derived from the primal.
The dual provides an alternative perspective on the problem.

Steps for Converting a Primal Problem to a Dual Problem (Non-Normal LP Problem)

Step 1: Convert the non-normal LP problem to normal form.
- Maximization Problems:
  - Inequalities:
    - '≤' type: No change required.
    - '≥' type: Multiply by -1; change to '≤'.
    - '=' type: Convert to two inequalities, one '≤' and one '≥'.
  - Unrestricted-in-sign variables: Rewrite as the difference of two non-negative variables.
- Minimization Problems:
  - Inequalities:
    - '≤' type: Multiply by -1; change to '≥'.
    - '≥' type: No change required.
    - '=' type: Convert to two inequalities, one '≥' and one '≤'.
  - Unrestricted-in-sign variables: Rewrite as the difference of two non-negative variables.

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Description

Explore the concepts of primal and dual forms in linear programming. Learn the steps to convert a non-normal LP problem to its dual form, including maximization and minimization scenarios. This quiz will enhance your understanding of alternative perspectives in linear problem-solving.