Determine an unbounded direction in a linear programming problem using simplex method tableau analysis | Step-by-Step Solution

Hello! I'm excited to help you understand how to find unbounded directions in linear programming - this is a really important concept that shows when a problem has no optimal solution because we can make the objective function infinitely large (or small)!

1. What We're Solving:

We need to identify when a linear programming problem is unbounded and construct a specific direction vector that proves the objective function can increase without limit while staying feasible.

2. The Approach:

In the simplex method, we detect unboundedness when we want to bring a variable into the basis (because it improves our objective), but there's no constraint limiting how large that variable can become.

WHY this matters: Understanding unbounded directions helps us recognize when a real-world problem might be poorly formulated or when we've made an error in our constraints.

3. Step-by-Step Solution:

Step 1: Recognize the Unbounded Signal

• You're in a simplex tableau and have identified a non-basic variable with a negative reduced cost (in maximization) or positive reduced cost (in minimization)
• When you look at that variable's column, ALL the coefficients are ≤ 0
• This means no constraint prevents this variable from increasing indefinitely!

Step 2: Construct the Unbounded Direction

• Start with the zero vector (all variables = 0)
• Set the entering variable (the one causing unboundedness) to 1
• For each basic variable, use the formula: direction component = -(corresponding tableau coefficient)
• Non-basic variables (except our entering one) stay at 0

Step 3: Verify Your Direction

• Check that A·d = 0 (where A is your constraint matrix and d is your direction)
• Check that c^T·d > 0 (where c is your objective coefficients) - this confirms the objective improves
• Verify that starting from any feasible point, you can move infinitely far in this direction while staying feasible

Step 4: Interpret the Result

• Any point of the form: x = x₀ + λd (where λ ≥ 0 and x₀ is feasible) remains feasible
• As λ → ∞, the objective value → ∞, proving unboundedness

4. The Framework:

For any unbounded LP problem, your analysis should include:

• The specific tableau that reveals unboundedness
• The constructed direction vector with clear calculations
• Verification that this direction satisfies the mathematical requirements
• Interpretation of what this means for the original problem

Key components of your direction vector:

• Entering variable coefficient: +1
• Basic variable coefficients: negative of their tableau entries
• Other non-basic variables: 0

5. Memory Tip:

Remember "No Limit Upward" - when you see Non-positive entries in a column with negative reduced cost, there's No Limit to going Upward! The direction vector points the way to infinity.

The key insight is recognizing that unboundedness isn't just about the math - it's about understanding when a system has no natural limits. Once you see that pattern in the tableau, constructing the direction becomes a systematic process.