When Excel Solver reports “The Set Objective Cell values do not converge” or “Solver found an unbounded solution”, it means the objective can increase (or decrease) without limit. This happens when the model is missing a constraint, has the wrong inequality direction, or allows variables to grow without bounds. This page shows how to diagnose unboundedness, how to fix it, and how to interpret Solver’s behavior using a complete numerical example.

• Linear Programming & Optimization
• Parent Business Analytics
• Inventory & Supply Chain Analytics
• Decision Analysis & Simulation

1. What “Unbounded” Means in Solver

A linear program is unbounded when the objective function can grow indefinitely while still satisfying all constraints. Solver stops because it cannot find a finite optimum.

Typical Solver messages:

• “The Set Objective Cell values do not converge.”
• “Solver found an unbounded solution.”
• “Objective keeps improving without limit.”

Unboundedness is almost always caused by a missing constraint or missing variable bounds.

2. Most Common Causes of Unbounded Models

2.1 Missing Upper or Lower Bounds

Solver allows variables to take extremely large positive or negative values unless you explicitly restrict them.

Example: Maximize \(Z = 40x\) with no constraints → Solver pushes \(x\) to infinity.

2.2 Wrong Inequality Direction

A “≥” that should be “≤” can open the feasible region in the direction of the objective.

2.3 Missing a Key Constraint

If a capacity, demand, or resource constraint is accidentally omitted, the model may allow unlimited production.

2.4 Allowing Negative Values When Not Intended

If a variable can go negative, the objective may improve without limit (especially in minimization problems).

3. Step‑by‑Step Process to Fix Unboundedness

3.1 Step 1 — Add Nonnegativity Constraints

• Set all decision variables to ≥ 0 unless negative values are explicitly allowed.

3.2 Step 2 — Check for Missing Constraints

• Verify that every resource, capacity, or requirement is included.
• Ensure no constraint was accidentally deleted or overwritten.

3.3 Step 3 — Verify Inequality Directions

• Check that “≤” constraints are not mistakenly written as “≥”.
• Ensure the feasible region is properly bounded.

3.4 Step 4 — Add Upper Bounds Where Needed

• Some models require explicit upper limits (e.g., demand limits, production caps).

3.5 Step 5 — Confirm the Objective Direction

• Ensure you selected Max or Min correctly.

4. Numerical Example: Fixing an Unbounded Solver Model

A company produces one product. Solver reports “unbounded.” Here is the original (incorrect) model.

4.1 Incorrect Model (Unbounded)

Decision variable

• \(x\): units of product to produce

Objective

Maximize: \[ Z = 50x \]

Incorrect constraints

• Machine time: \[ 2x \le 200 \] This is correct.
• Missing nonnegativity: Solver allows \(x\) to be negative.
• Missing demand or sales limit: Nothing prevents \(x\) from increasing indefinitely.

Because the objective increases with \(x\), and there is no upper bound other than machine time, Solver pushes \(x\) to 100 (the machine limit). But if the machine constraint is accidentally removed, Solver pushes \(x\) to infinity → unbounded.

5. Fixing the Unbounded Model

5.1 Add Nonnegativity

\[ x \ge 0 \]

5.2 Add a Demand Limit (If Required)

\[ x \le 120 \]

5.3 Ensure Machine Constraint Is Present

\[ 2x \le 200 \]

With these constraints, the feasible region is now bounded.

6. Solving the Corrected Model

The feasible region is defined by:

• \(x \ge 0\)
• \(x \le 120\)
• \(2x \le 200 \Rightarrow x \le 100\)

The tightest upper bound is \(x = 100\).

Optimal solution:

• \(x = 100\)
• Profit: \[ Z = 50(100) = 5000 \]

Solver now finds a finite optimum because the model is properly bounded.

7. Additional Tips for Preventing Unboundedness

• Always add nonnegativity constraints for LP variables.
• Check that every resource has a corresponding constraint.
• Ensure demand or sales limits are included when appropriate.
• Use “Reset All” in Solver if the model becomes corrupted.

For more help with LP, Solver, and optimization modeling, explore:

• Linear Programming & Optimization
• Parent Business Analytics
• Inventory & Supply Chain Analytics
• Decision Analysis & Simulation