$1.$ Formulate the following as a linear programming problem:

A greenhouse operator plans to bid for the job of providing flowers for city parks. He will use tulips, daffodils, and flowering shrubs in three types of layouts. A Type $1$ layout uses $40$ tulips, $30$ daffodils, and $3$ flowering shrubs. A Type $2$ layout uses $20$ tulips, $35$ daffodils, and $4$ flowering shrubs. A Type $3$ layout uses $10$ tulips, $60$ daffodils, and $2$ flowering shrubs. The net profit is $ $45$ for each Type $1$ layout, $$35$ for each Type $2$ layout, and $$55$ for each Type $3$ layout. He has $1,100$ tulips, $900$ daffodils, and $80$ flowering shrubs. How many layouts of each type should be used to yield maximum profit?

$2.$ Explain how your formulation and solution in the above problem changes if both of the following conditions are added:

a$.$ The number of Type $1$ layouts cannot exceed the number of Type $2$ layouts.

b$.$ There must be at least four layouts of each type.