A factory makes two products — formulate and solve the LP problem graphically

Question

A factory manufactures two products A and B. Product A requires 2 hours of machine time and 1 hour of labour. Product B requires 1 hour of machine time and 2 hours of labour. Available machine time is 100 hours and labour time is 80 hours per week. Profit on A is ₹50 and on B is ₹40. Formulate and solve the LP problem graphically to maximize profit.

Solution — Step by Step

Let $x$ = number of units of Product A produced per week

Let $y$ = number of units of Product B produced per week

Since we cannot produce negative quantities: $x \geq 0$ , $y \geq 0$ (non-negativity constraints)

Machine time constraint: $2x + y \leq 100$ (A uses 2 hours, B uses 1 hour; max 100 hours)

Labour constraint: $x + 2y \leq 80$ (A uses 1 hour, B uses 2 hours; max 80 hours)

Objective function (maximize profit): $Z = 50x + 40y$

The feasible region is defined by all four constraints. Find the vertices (corner points):

Corner 1: Origin $(0, 0)$

Corner 2: $x$ -intercept of machine constraint: $2x = 100 \Rightarrow x = 50$ . Point: $(50, 0)$

Corner 3: $y$ -intercept of labour constraint: $2y = 80 \Rightarrow y = 40$ . Point: $(0, 40)$

Corner 4 (intersection of the two lines):

$2x + y = 100$ … (i)

$x + 2y = 80$ … (ii)

From (i): $y = 100 - 2x$ . Substitute in (ii): $x + 2(100 - 2x) = 80$

$x + 200 - 4x = 80 \Rightarrow -3x = -120 \Rightarrow x = 40$

$y = 100 - 80 = 20$ . Point: $(40, 20)$

Corner point	$Z = 50x + 40y$
$(0, 0)$	$0$
$(50, 0)$	$50 \times 50 = 2500$
$(0, 40)$	$40 \times 40 = 1600$
$(40, 20)$	$50 \times 40 + 40 \times 20 = 2000 + 800 = 2800$

Maximum $Z = \text{₹}2800$ at $(40, 20)$

Why This Works

The fundamental theorem of linear programming: the maximum (or minimum) of a linear objective function over a convex feasible region always occurs at a corner (vertex) of the region. This is because a linear function increases uniformly in one direction — it will be maximized at the extreme boundary, which for a polygon is always a corner.

We never need to check interior points — only the finitely many corners.

Answer: Produce 40 units of A and 20 units of B per week for maximum weekly profit of ₹2800.

Common Mistake

Students sometimes forget to check ALL corner points, or they include only the intercepts of one constraint. The optimal solution frequently occurs at the intersection of two constraint lines (like $(40, 20)$ here), not at an axis intercept. Always find all corners: origin, both axis intercepts, and ALL intersection points of pairs of constraint lines that fall within the feasible region.

Question

Solution — Step by Step

Why This Works

Common Mistake

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