Q.A manufacturer of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs 50 and that on type B circuit is Rs 60, formulate this problem as a LPP so that the manufacturer can maximise his profit.

To formulate the LPP, translate the stock limits and the profit into linear inequalities and a linear objective.

Decision variables. Let $x$ = number of type A circuits, $y$ = number of type B circuits.

Constraints (stock limits).

• Resistors: $20x + 10y \le 200$
• Transistors: $10x + 20y \le 120$
• Capacitors: $10x + 30y \le 150$
• Non-negativity: $x \ge 0$, $y \ge 0$ …

Why this is a Linear Programming Problem

The stock of resistors, transistors and capacitors is limited, and the two circuits use these parts in fixed amounts while each earns a fixed profit. Deciding how many of each circuit to make so that no stock is exceeded and the profit is greatest is exactly a linear programming problem (LPP): a linear objective to maximise, subject to linear inequality constraints and non-negativity conditions.

Step-by-step formulation

1. Decision variables.

Let $x$ = number of type A circuits and $y$ = number of type B circuits, with $x \ge 0$, $y \ge 0$.

2. Resource constraints.

Type A uses 20 resistors, 10 transistors, 10 capacitors; type B uses 10 resistors, 20 transistors, 30 capacitors. Stock available: 200 resistors, 120 transistors, 150 capacitors.

• Resistors: $20x + 10y \le 200$
• Transistors: $10x + 20y \le 120$
• Capacitors: $10x + 30y \le 150$

3. Objective function. …

Method: Formulating an LPP from a Word Problem

Use this to translate a resource-allocation story into a linear programming problem (decision variables, constraints, objective) without solving it.

Steps

Step 1: Define the decision variables.

Name what the decision-maker chooses — e.g. let $x$ and $y$ be the numbers of each product to make. State them clearly; every later expression is built from these.

Step 2: Write one constraint per limited resource.

For each resource, the amount consumed must not exceed the stock available:

$(\text{use per unit of }x)\,x + (\text{use per unit of }y)\,y \;\le\; (\text{available}).$ …

Common Mistakes

Mistake 1: Swapping the resource-to-product coefficients.

Why it's wrong: type A uses 20 resistors, so the resistor constraint is $20x+10y\le200$, not $10x+20y\le200$. Mismatching a rate to the wrong variable corrupts the whole constraint. Correct approach: for each resource, multiply each product's own per-unit usage by its variable.

Mistake 2: Getting the inequality direction wrong. …