Q.A dietician wishes to mix two types of foods in such a way that the mixture contain at least 8 units of Vitamin A and 10 units of Vitamin C. Food-I contains 2 units/kg of Vitamin A and 1 unit/kg of Vitamin C, while Food-II contains 1 unit/kg of Vitamin A and 2 units/kg of Vitamin C. It costs Rs. 50 per kg to purchase Food-I and Rs. 70 per kg to purchase Food-II. Formulate the above as a LPP to minimise the cost of such a mixture.

Identify the decision variables, the profit function to maximize, and translate each stated restriction into an inequality.

Decision variables: let $x$ = number of sheep purchased, $y$ = number of goats purchased.

Objective function: the farmer earns profit $\text{Rs.}\,150$ per sheep and $\text{Rs.}\,200$ per goat, so total profit:

$Z = 150x+200y \quad(\text{maximize})$

Budget constraint: sheep cost Rs. 1,500 each and goats Rs. 2,000 each; total spend can't exceed Rs. 60,000:

$1500x+2000y\le60000 \;\Rightarrow\; 3x+4y\le120 \quad(\text{dividing by }500)$

Capacity constraint: he cannot accommodate more than 100 animals:

$x+y\le100$

…

Define decision variables for units of each food, write the cost as the objective, and translate each nutrient requirement into a linear inequality.

Let $x$ = number of units of food A, and $y$ = number of units of food B.

Objective (minimize cost): Food A costs Rs.,4/unit, food B costs Rs.,3/unit, so:

$\text{Minimize } Z = 4x+3y$

Constraints (from nutrient requirements):

Vitamins: A gives $200$ units/unit, B gives $100$ units/unit, at least $4000$ needed:

$200x+100y \ge 4000 \ \Rightarrow\ 2x+y\ge40$

Minerals: A gives $1$ unit/unit, B gives $2$ units/unit, at least $50$ needed:

$x+2y\ge50$

Calories: A gives $40$/unit, B gives $40$/unit, at least $1400$ needed: …

Translate each real-world limit (labour-hours available for fabricating, and for finishing) into a linear inequality, and the profit goal into a linear objective — that is the entire LPP formulation.

Step 1 — define decision variables. Let $x=$ number of units of model $A$ produced, $y=$ number of units of model $B$ produced. Both must be non-negative: $x\ge0,\ y\ge0$.

Step 2 — fabricating-hours constraint. Model $A$ needs $9$ labour-hours, model $B$ needs $12$, and only $180$ hours are available:

$9x+12y\le180.$

Step 3 — finishing-hours constraint. Model $A$ needs $1$ hour, model $B$ needs $3$, and only $30$ hours are available: …

Translate each sentence of the word problem into a linear inequality: decision variables = kg of each food; objective = total cost; constraints = the minimum vitamin requirements.

Step 1 — define decision variables.

Let $x$ = kg of Food-I used, $y$ = kg of Food-II used.

Step 2 — write the objective function (minimise total cost). Food-I costs Rs.,50/kg, Food-II costs Rs.,70/kg:

$Z = 50x + 70y \quad (\text{to be minimised})$

Step 3 — write the constraints.

Vitamin A requirement: Food-I supplies $2$ units/kg, Food-II supplies $1$ unit/kg, and the mixture must have at least $8$ units:

$2x + y \ge 8$

…

Translate labour and profit data into decision variables, an objective function, and constraints.

Let $x$ = number of units of model A produced, $y$ = number of units of model B produced.

Objective (profit to maximize): Model A gives ₹8,000 profit/unit, model B gives ₹12,000 profit/unit:

$\text{Maximize } Z=8000x+12000y$

Constraints:

• Fabricating hours: model A needs 9 hrs/unit, model B needs 12 hrs/unit, available 180 hrs: $9x+12y\le180$
• Finishing hours: model A needs 1 hr/unit, model B needs 3 hrs/unit, available 30 hrs: $x+3y\le30$ …