1. What Is Operations Research?

Operations research (OR) is the discipline of making decisions with mathematical models. The phrase is old — it comes from the analysis groups set up in Britain and the United States during the Second World War to study radar deployment, convoy sizing and bombing patterns — but the activity it names is now everywhere: in the software that routes parcel vans, sets airline fares, schedules nurses, and decides which power stations run tomorrow morning.

These notes are about those applications. Each lecture picks one decision problem that organizations actually face and works through it end to end.

The modelling cycle

Every application in these notes follows the same loop.

Describe the decision. Who decides what, how often, and under which rules? A depot manager choosing tomorrow's van routes faces a different problem from a planner choosing where to build depots for the next ten years, even though both are "logistics".
Formulate a model. Name the decision variables (what you control), the objective (what you are trying to minimize or maximize), and the constraints (what must hold). The output is a precise mathematical statement.
Solve it. Choose an algorithm — exact, heuristic, or simulation — appropriate to the model's structure and size.
Interpret and validate. Check that the answer makes sense to the people who will use it. Ask what the model left out. Usually you return to step 1.

(Optimization model)

An optimization model has the form

\min_{x \in X} \; f(x) \quad \text{subject to} \quad g_i(x) \le 0, \; i = 1, \dots, m,

where $x$ is the vector of decision variables, $X$ is its domain (for example $\mathbb{R}^n_{\ge 0}$ or $\mathbb{Z}^n$ ), $f$ is the objective, and the $g_i$ are constraints. A point satisfying every constraint is feasible; a feasible point with the smallest objective value is optimal.

Step 2 is where most of the craft lies. The same real problem can usually be written in several mathematically equivalent ways, and they are not equally easy to solve. You will see this repeatedly — for example in the "weak" and "strong" formulations of facility location.

The toolkit

OR draws on a fairly small set of model classes. Most applications in these notes use one or two of them.

Model class	Variables	Typical applications	Workhorse methods
Linear programming (LP)	continuous	transportation, blending, production planning	simplex, interior point
Integer / mixed-integer programming (MIP)	some integer or binary	facility location, scheduling, routing	branch and bound, cutting planes
Network flows	flows on arcs	transportation, assignment, shortest paths	specialized simplex, combinatorial algorithms
Nonlinear / quadratic programming	continuous	portfolio selection, pricing	interior point, gradient methods
Stochastic models	random parameters	inventory, revenue management	newsvendor analysis, dynamic programming, sampling
Queueing theory	arrival and service processes	call centres, hospitals, networks	closed-form formulas, simulation
Heuristics and metaheuristics	any	large routing and scheduling instances	construction, local search, tabu, genetic algorithms

Linear programming deserves special mention because it underpins much of the rest. Branch and bound for integer programs solves a sequence of LPs; column generation solves LPs with too many variables to write down; and the dual of an LP gives prices — shadow prices of resources — that are often as useful to a decision-maker as the solution itself.

Linear programming in one page

Since LP recurs throughout, here is the minimum we will rely on.

A linear program in standard inequality form is

\min_{x} \; c^\top x \quad \text{s.t.} \quad Ax \ge b, \; x \ge 0 .

Its dual is

\max_{y} \; b^\top y \quad \text{s.t.} \quad A^\top y \le c, \; y \ge 0 .

(LP duality)

If either problem has a finite optimum, so does the other, and the optimal values are equal: $c^\top x^* = b^\top y^*$ . Moreover $x^*$ and $y^*$ satisfy complementary slackness: $y_i^* > 0$ only if constraint $i$ is tight, and $x_j^* > 0$ only if dual constraint $j$ is tight.

The economic reading is what makes duality useful in applications. The dual variable $y_i^*$ is the rate at which the optimal cost changes as the right-hand side $b_i$ changes — the shadow price of constraint $i$ . If constraint $i$ says "meet demand at market $i$ ", then $y_i^*$ is the marginal cost of serving one more unit there. We will use this interpretation in the transportation problem, in electricity markets (where shadow prices are the market prices), and in airline revenue management (where they become bid prices).

How these notes are organized

The lectures are grouped by application area rather than by technique, because that is how problems arrive in practice: nobody hands you "an integer program", they hand you a depot and a fleet of vans. Each lecture follows roughly the same structure:

The decision — the setting and what is being chosen.
Formulation — the model, with every variable and constraint explained.
Solution methods — how the model is solved, and what makes it easy or hard.
A worked example — small enough to check by hand.
Extensions — what real deployments add on top.

Use the panel on the left to move between lectures; within a lecture it also lists the sections.