"Holds up to 40 pieces" is a maximum-capacity condition, giving the constraint $x \leq 40$.

Let $x$ be the number of pieces of large crockery loaded into the dish-washing machine.

The phrase "holds up to 40 pieces" means the machine can take a maximum of 40 pieces — i.e. $x$ can be anything from $0$ up to $40$, but not more than $40$. A maximum/ceiling condition is written as a $\leq$ inequality:

$x \leq 40$

…

The objective function of an LPP is the linear function of the decision variables that is to be optimised (maximised or minimised), such as $Z=ax+by$ — option (b).

Every linear programming problem has three parts:

• Decision variables — the unknown quantities to be determined (e.g. $x, y$).
• Objective function — a linear expression such as $Z=ax+by$ that measures the quantity to be optimised; the aim is to maximise it (e.g. profit) or minimise it (e.g. cost).
• Constraints — the linear inequalities/equations that limit the values of the decision variables, together with the non-negativity restrictions. …