Linear Programming · Study notes

MATFormulation

Formulation of a Linear Programming Problem

Formulation is the modelling step: converting a word problem into a precise mathematical LPP without solving it. You identify decision variables, write the linear objective function to optimise, and express all resource limits and requirements as linear inequalities together with non-negativity.

ISC awards marks specifically for a clean, correctly-labelled formulation, so accuracy of variables, units, and inequality direction is essential.

Objective function

Z = ax + by \quad (\text{maximise profit or minimise cost})

x,y

are decision variables (quantities to decide);

a,b

are per-unit profit or cost coefficients.

Resource (capacity) constraint

a_1 x + b_1 y \le c_1

a_1,b_1

are the resource used per unit of

x,y

;

c_1

is the available amount ("at most").

Requirement (demand) constraint

a_2 x + b_2 y \ge c_2

a_2,b_2

are the nutrient/output supplied per unit;

c_2

is the minimum needed ("at least").

Non-negativity restriction

x \ge 0, \qquad y \ge 0

Quantities of products, foods, or loads cannot be negative; always included in every LPP.

Step 1: Let $x$ and $y$ denote the unknown quantities, stating exactly what each represents and its unit.
Step 2: Write the linear objective $Z=ax+by$ and say whether it is to be maximised or minimised.
Step 3: For each resource, sum the per-unit usage times the variable and bound it by availability with $\le$ .
Step 4: For each requirement, write the supplied-amount expression as $\ge$ the minimum needed.
Step 5: Append the non-negativity constraints $x\ge0,\ y\ge0$ .
All relations must be LINEAR in $x,y$ (no products $xy$ , powers, or ratios) for it to be a valid LPP.
When the question says "formulate (do not solve)", stop after writing the model; full marks come from the correct system, not a graph.
Keep the table of per-unit data straight: rows are products/foods, columns are resources/nutrients, to avoid swapping coefficients.

Where the marks go

Defining variables vaguely ("let $x$ be product A") instead of "let $x$ be the number of units of A produced per day".
Using the wrong inequality direction: capacity should be $\le$ but a requirement should be $\ge$ .
Transposing the per-unit coefficients between the two products, so the constraint coefficients are mismatched.
Forgetting the non-negativity constraints $x\ge0,\ y\ge0$ , which costs a formulation mark.

How the board asks it

Applicationthe full 5-step formulation method
A manufacturer makes two products $A$ and $B$ . Each unit of $A$ requires $3$ hours on machine $M_1$ and $1$ hour on machine $M_2$ , while each unit of $B$ requires $2$ hours on $M_1$ and $4$ hours on $M_2$ . Machines $M_1$ and $M_2$ are available for at most $42$ and $48$ hours respectively. The profit is Rs. $50$ on each unit of $A$ and Rs. $60$ on each unit of $B$ . Formulate this as an LPP to maximise the profit (do not solve).
Applicationrequirement constraints with $\ge$
A diet for a sick person must contain at least $4000$ units of vitamins, $50$ units of minerals and $1400$ calories. Two foods $X$ and $Y$ are available: each unit of $X$ supplies $200$ units of vitamins, $1$ unit of minerals and $40$ calories, while each unit of $Y$ supplies $100$ units of vitamins, $2$ units of minerals and $40$ calories. If $X$ costs Rs. $4$ and $Y$ costs Rs. $3$ per unit, formulate the LPP to minimise the cost of the diet (do not solve).
Numericalidentifying decision variables and objective function
A farmer plans to plant $x$ acres of wheat and $y$ acres of rice. Wheat yields a profit of Rs. $11000$ per acre and rice Rs. $7000$ per acre. State the decision variables together with their units, and write the objective function $Z$ that the farmer wishes to maximise.
Give reasonsthe linearity condition for a valid LPP
While modelling a manufacturing problem a student writes the constraint $xy \le 20$ . State, giving a reason, whether the resulting system can be a valid linear programming problem.
Multiple choiceinequality direction for a capacity constraint
A transport company has trucks that can carry a load of at most $200$ quintals, where $x$ and $y$ are the quantities of two goods loaded. The constraint representing this capacity is: (a) $x+y \ge 200$ (b) $x+y \le 200$ (c) $x+y = 200$ (d) $x+y \le 0$ . Choose the correct option.

Practise this topic

NextGraphical Method & the Feasible Region

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.