Q.A firm manufactures two products A and B. Each unit of A requires 2 hours on machine I and 1 hour on machine II; each unit of B requires 1 hour on machine I and 3 hours on machine II. Machine I is available for 8 hours and machine II for 9 hours per day. The profit is ₹5 per unit of A and ₹4 per unit of B. Formulate this as an LPP to maximise the daily profit.

"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:

$x \leq 40$

Together with the non-negativity restriction $x \geq 0$ (a count of pieces cannot be negative), this is the required constraint.

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.

The objective function is therefore the function to be maximised or minimised. A constraint restricts the variables, the set of points satisfying all constraints is the feasible region, and an equation of a straight line is only one boundary — none of these is the objective function.