Mathematical Formulation of an LPP

First choose variables with units. Next identify whether the objective is maximum or minimum. Then convert each condition into an inequality or equation. Finally add x >= 0 and y >= 0 wherever negative values are impossible.

A diagram is not compulsory for formulation alone, but the formulated inequalities must be suitable for drawing half-planes in the graphical method.

Writing x + y >= 100 for a statement like storage can hold at most 100 packs is a realistic wrong step.

A linear programming problem is formulated by choosing decision variables, writing a linear objective function such as Z = ax + by, and expressing all given conditions as linear constraints.

A linear programming problem is formulated by choosing decision variables, writing a linear objective function such as Z = ax + by, and expressing all given conditions as linear constraints. General form for Class 12 graphical LPP: maximize or minimize Z = ax + by, subject to constraints a1x + b1y <=, >=, or = c1; a2x + b2y <=, >=, or = c2; and usually x >= 0, y >= 0. Use <= for upper limits and >= for minimum requirements. If x chairs and y tables are produced, and profit is Rs 30 per chair and Rs 50 per table, then the objective function is Z = 30x + 50y. If each chair needs 2 hours and each table needs 4 hours, with at most 40 hours available, then 2x + 4y <= 40.

Formulation converts a word problem into mathematics. The unknown quantities are represented by variables, the quantity to be maximized or minimized becomes the objective function, and each limitation or requirement becomes a linear inequality. Non-negativity conditions such as x >= 0 and y >= 0 are included when variables represent quantities like units, hours, kilograms, or packets. General form for Class 12 graphical LPP: maximize or minimize Z = ax + by, subject to constraints a1x + b1y <=, >=, or = c1; a2x + b2y <=, >=, or = c2; and usually x >= 0, y >= 0. Use <= for upper limits and >= for minimum requirements. A shopkeeper sells x packs of type A and y packs of type B. Profit is Rs 20 on A and Rs 30 on B. Storage allows at most 100 packs, and type B packs must be at least 20. Formulate the LPP. Objective: maximize Z = 20x + 30y. Storage constraint: x + y <= 100. Minimum B condition: y >= 20. Non-negativity: x >= 0, y >= 0. Therefore, maximize Z = 20x + 30y subject to x + y <= 100, y >= 20, x >= 0, y >= 0. Usually asked as a short-answer or long-answer first part: define variables, write the objective function, list constraints, and mention non-negativity restrictions. Writing x + y >= 100 for a statement like storage can hold at most 100 packs is a realistic wrong step.