Lesson 1 of 9 · 6 min
Optimisation problems
NCERT §12.1
The science club runs a bake stall at the school fete. With one oven and one sack of flour, how many trays of cupcakes and of cookies should they bake to make the most money?
The story this chapter follows: The fete bake stall
The lesson in notes
In short
An optimisation problem asks for the largest profit, the smallest cost, or the least use of some resource, among all the choices that the situation allows.
The chapter's running example: a furniture dealer stocks only tables and chairs, has Rs 50,000 to spend and room for 60 pieces in all. Each table costs Rs 2500 and earns Rs 250; each chair costs Rs 500 and earns Rs 75.
Trying plans by hand: tables alone means 50000 ÷ 2500 = 20 tables and Rs 5000 profit. Chairs alone would allow 100 chairs by money, but only 60 fit, giving Rs 4500. A mix of 10 tables and 50 chairs gives Rs 6250.
Different plans give different profits, and there are infinitely many plans in between, so guessing cannot prove which is best. Linear programming answers this systematically.
When the profit (or cost) and every limit are linear in the unknowns, the optimisation problem is a linear programming problem. This chapter solves such problems in two variables by the graphical method only.