If the number of airplanes purchased by HelloAIR must have an integer value, which assumption has been violated?

In this scenario, the assumption that has been violated is the integrity of the decision variable. In a typical optimization problem, particularly in linear programming, decision variables must be defined in a manner that respects the constraints of the problem being analyzed.

Explanation

1. Integer Constraint: The question indicates that the number of airplanes purchased must be an integer value. This introduces an integer constraint to the problem. If the model or assumption allows for non-integer values (such as fractions or decimals), this assumption is violated.

2. Decision Variables: If the problem is structured in such a way that it allows for any real number, it goes against the practical scenario of purchasing whole airplanes, which cannot be done in fractions.

3. Practical Application: In scenarios such as this, it's crucial to model real-world situations accurately. The decision variable representing the number of airplanes should only take on non-negative integer values.

Summary

In conclusion, the violation lies in the failure to properly define the decision variables as integers within the model, meaning that the purchasing assumption did not reflect real-world constraints.