Fuel blending problem

Imagine you run a refinery. You have 3 raw materials stored in tanks — each with a limited quantity and a different cost. By mixing these raw materials in different proportions you can produce 3 types of fuel (STAR-98, Unleaded, and Super), each with a different selling price.

The problem is that not any blend will do: each fuel type has strict rules. For example, STAR-98 requires at least 40% of the blend to be Raw Material 2 — which happens to be the most expensive. Super requires at least 70% of Raw Material 1, the cheapest.

The question to answer: how much of each fuel type to produce to maximize total profit, without exceeding available quantities of each raw material and meeting all composition requirements?

Available raw materials

Raw Material 1

The cheapest

3,000 u.

$3/u

Raw Material 2

The most expensive · critical resource

2,000 u.

$6/u

Raw Material 3

Intermediate price

4,000 u.

$4/u

Fuel types to produce

A · STAR-98$5.5/u

MP1 ≤ 30% · MP2 ≥ 40% · MP3 ≤ 50%

Cheapest possible blend: 30% RM1 · 40% RM2 · 30% RM3 → net margin ~$1.0/u

B · Unleaded$4.5/u

MP1 ≤ 50% · MP2 ≥ 10%

Cheapest possible blend: 50% RM1 · 10% RM2 · 40% RM3 → net margin ~$0.8/u

C · Super$3.5/u

MP1 ≥ 70%

Cheapest possible blend: 100% RM1 → net margin ~$0.5/u

The model has 9 blend variables + production and usage variables = over 15 variables and 9 constraints on proportions and availability — impossible to solve by hand.