. What does it mean to say that a linear programming problem

(a) Is in standard form?

(b) Is in canonical form?

(c) has an unbounded solution?

(d) has a degenerate solution?

(e) has no feasible solution?

In linear programming, several specific terms are used to describe the structure and characteristics of a problem. Here's what each term means:

### (a) **Standard Form**

A linear programming problem is in **standard form** when:
- The objective is to **maximize** (or minimize) a linear function.
- All the **constraints** are expressed as **linear equalities**.
- All the **variables** are **non-negative**.

Mathematically, it looks like this:

\[
\text{Maximize } c^T x
\]
\[
\text{Subject to: } A x = b, \quad x \geq 0
\]

Where:
- \( x \) is the vector of variables,
- \( c^T x \) is the objective function (to be maximized),
- \( A x = b \) are the equality constraints,
- \( x \geq 0 \) means all variables are non-negative.

---

### (b) **Canonical Form**

A linear programming problem is in **canonical form** when:
- The objective is to **maximize** a linear function.
- The constraints are expressed as a mix of **linear inequalities** (usually in the form of \(\leq\)) and **non-negative variables**.

Mathematically, it looks like this:

\[
\text{Maximize } c^T x
\]
\[
\text{Subject to: } A x \leq b, \quad x \geq 0
\]

Where:
- \( A x \leq b \) are inequality constraints.
  
In contrast to standard form, canonical form allows inequalities instead of equalities.

---

### (c) **Unbounded Solution**

A linear programming problem has an **unbounded solution** when:
- The objective function can increase (for maximization) or decrease (for minimization) without bound.
- This means there is **no finite optimal value** because the objective function can go to infinity (or negative infinity) while still satisfying the constraints.

An unbounded solution typically happens when the feasible region (the set of points satisfying all constraints) is open and extends indefinitely in a direction that improves the objective function.

---

### (d) **Degenerate Solution**

A linear programming problem has a **degenerate solution** when:
- At least one of the basic variables in the solution has a value of **zero**.
- Degeneracy can occur when there are **multiple optimal solutions** or when the problem's constraints are such that one or more variables don't need to take positive values in the solution.

Degeneracy can cause cycling in some algorithms like the Simplex Method, though modern implementations handle this effectively.

---

### (e) **No Feasible Solution**

A linear programming problem has **no feasible solution** when:
- There is no set of values for the decision variables that simultaneously satisfy all of the constraints.
- This can happen when the constraints are **inconsistent** or contradict each other, leading to an **empty feasible region** (i.e., no solution exists that satisfies all the constraints).

---

### Want more details or have any questions?
Here are 5 related questions you might explore next:
1. How does the Simplex Method handle unbounded solutions?
2. Can a degenerate solution still be optimal, and why?
3. How can you transform a linear programming problem into standard form?
4. What are the consequences of having no feasible solution in practical problems?
5. How can degeneracy affect the performance of linear programming algorithms?

**Tip:** In practice, always check for feasibility and boundedness before trying to solve a linear programming problem—it helps avoid wasted computation time!

What does it mean to say that a linear programming problem (a) is in standard form? (b) is in canonical form? (c) has an unbounded solution? (d) has a degenerate solution? (e) has no feasible solution?

Explore the definitions and characteristics of linear programming problems, including standard form, canonical form, unbounded solutions, degenerate solutions, and no feasible solutions. Understand how these concepts impact the optimization process with examples and key formulas. Suitable for students studying operations research or optimization at the undergraduate level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation