Solve the blending problem in Python

The blending problem — mix raw components into a product that meets a quality specification at minimum cost — is a classic linear program: fuel blending, feed and food formulation, alloy mixing. With quicopt it is a few lines in Python.

The naive approach

Trial-and-error on the mix ratios can find a blend that meets spec, but not the cheapest one, and it gets unwieldy with more components and specs. When quality is a volume-weighted average, the problem is linear — so solve it exactly.

Model it as an LP

A continuous variable per component is how much to use. One constraint fixes the total volume, another enforces the minimum quality (a volume-weighted average), and the objective minimizes cost:

from ortools.math_opt.python import mathopt
from quicopt import Client

# Blend components into 100 units of product at minimum cost, meeting a minimum
# quality (octane) spec.
octane = [95.0, 88.0, 92.0, 85.0]
cost   = [6.0, 4.5, 5.5, 3.8]
N = len(octane)

model = mathopt.Model(name="blending")
x = [model.add_variable(lb=0.0, name=f"comp_{i}") for i in range(N)]
model.add_linear_constraint(sum(x) == 100.0)
model.add_linear_constraint(sum(octane[i] * x[i] for i in range(N)) >= 90.0 * 100.0)
model.minimize(sum(cost[i] * x[i] for i in range(N)))

client = Client("https://try.quicoptapi.pgi.fz-juelich.de")
print(client.solve(model).display)

├── status:     optimal
├── feasible:   true
├── objective:  489.99999999999983
├── x:          comp_0=50.0, comp_1=0, comp_2=0, comp_3=50.0  (4 variables)
└── solve_time: 0.0022 s

What you get

status: optimal means the blend is proven cheapest — cost 490 (the tiny …9983 tail is ordinary floating point), mixing 50 units each of the highest- and lowest-octane components to hit exactly the 90 spec at least cost.

The same model scales to many components and several specs at once — you change the data, not the method. (If streams mix in shared tanks, quality stops being linear and you are in non-convex NLP territory — which quicopt also solves.)

Next

• The problem class behind this: Linear programming (LP)
• A runnable model for every supported class: Examples
• Set up the client and solve your first model: Getting started

Reference: H. P. Williams, Model Building in Mathematical Programming, Wiley (5th ed., 2013).