Q.A manufacture produces bulbs and tubes. Each of these must be processed through two machines M1 and M2. A package of bulbs require 1 hour of work on Machine M1 and 3 hours of work on M2. A package of tubes require 2 hours on Machine M1 and 4 hours on Machine M2. He earns a profit of Rs. 13.5 per package of bulbs and Rs. 55 per package of tubes. Formulate the L.P.P. to maximize the profit, if he operates the machine M1, for atmost 10 hours a day and machine M2 for atmost 12 hours a day.

Formulating an L.P.P. means translating a real-world word problem into the precise mathematical shape of decision variables, an objective function, and a set of constraints. The chapter gives a fixed three-step working rule: Step 1 — identify the decision variables and assign them symbols such as $x,y$ (or $x_1,x_2$), remembering to introduce the non-negativity constraints $x\ge0,y\ge0$; Step 2 — identify what is to be optimized (usually total profit or total cost) and write it as a linear expression in the decision variables — this is the objective function; Step 3 — identify every resource limit or requirement stated in the problem (machine-hours available, raw material availability, minimum nutrient needs, budget, storage space, etc.) and express each one as a linear inequation. A resource that is available only up to some maximum amount gives a $\le$ constraint (e.g. machine-hours, budget, storage space); a requirement that must be met at least to some minimum level gives a $\ge$ constraint (e.g. minimum nutrients in a die …