Operations research and logistics

Level UniversityDifficulty ★★★★★Application⌖ Open in the map

What is it?

Optimal decisions under constraints: production plans, delivery routes, crew schedules, network flows. Continuous problems are solved with convex optimization and duality (Lagrange multipliers as shadow prices); discrete ones add integer programming on top.

Formulas

$\min_x\ c^{\mathsf T}x \quad\text{subject to}\quad Ax \le b,\ x \ge 0$
a linear program

Why does it matter?

Airlines, logistics companies, energy grids and cloud schedulers solve such problems continuously; calculus supplies optimality conditions and sensitivities.

The mathematics behind it

Convexity and concavity★★★★★fundamental
Convex optimization (linear, quadratic, conic programs) is the workhorse of planning and logistics.
Constrained optimization★★★★★fundamental
Scheduling, routing and allocation are optimization under capacity and demand constraints.
KKT conditions★★★★★fundamental
Interior-point and active-set solvers are algorithms for satisfying the KKT conditions.
Maxima and minima★★★★★fundamental
Cost minimization and profit maximization are the native language of operations research.
Lagrange multipliers★★★★★frequent
Dual variables are shadow prices: how much an extra unit of a resource is worth.
Weierstrass extreme value theorem★★★★★advanced
Existence of an optimal solution: a continuous cost on a compact feasible set always has a minimum.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.