Linear programming: formulate and solve graphically

Linear programming

A-level Further Maths (9FM0) · Decision 1

ORIGINAL

Define the variables in words, write the objective and constraints, then test the vertices of the feasible region.

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Question

A carpenter makes chairs and tables. Each chair needs 2 hours of cutting and 2 hours of finishing; each table needs 3 hours of cutting and 1 hour of finishing. Each week at most 24 hours of cutting and 16 hours of finishing are available. The profit is £30 per chair and £40 per table. Formulate this as a linear programming problem and use a graphical method to find how many chairs and tables should be made each week to maximise the profit.

Every step worked, with the reasoning.

1
Let $x$ be the number of chairs and $y$ the number of tables made each week.
Define the decision variables in words, including the time period.
2
Maximise $P = 30x + 40y$
The objective function: total weekly profit in pounds.
3
Subject to $2x + 3y \le 24$ , $\ 2x + y \le 16$ , $\ x \ge 0$ , $\ y \ge 0$
Cutting hours, finishing hours and non-negativity.
4
Draw $2x + 3y = 24$ through $(12, 0)$ and $(0, 8)$ , and $2x + y = 16$ through $(8, 0)$ and $(0, 16)$ . The feasible region $R$ lies on or below both lines with $x, y \ge 0$ .
Each $\le$ constraint keeps the side of its line containing the origin.
5
Vertices of $R$ : $(0, 0)$ , $(8, 0)$ , $(0, 8)$ and the intersection: subtracting the equations gives $2y = 8$ , so $y = 4$ , $x = 6$ , i.e. $(6, 4)$ .
Solve the two constraint lines simultaneously for the corner they share.
6
$P(8, 0) = 240$ , $\ P(0, 8) = 320$ , $\ P(6, 4) = 180 + 160 = 340$
Vertex method: a linear objective is optimised at a vertex. (Equivalently, slide the objective line $30x + 40y = k$ away from the origin — the last point of $R$ it touches is $(6, 4)$ .) The optimum is already integer-valued.

Answer: Make 6 chairs and 4 tables each week, for a maximum profit of £340.

Question

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