In 1940, with bombers overhead, Britain put mathematicians to work on questions no equation had ever been asked: how big should a convoy be, where should radar go, how deep should a depth charge explode? The answers won battles — and a new discipline was born. Operations research (OR) is the science of making the best decision when resources are scarce, choices are many and the numbers are against you. It is applied mathematics pointed at the running of the real world: airlines routing planes, hospitals scheduling theatres, Amazon stocking warehouses, a factory deciding what to build.
This masters-level course builds the whole toolkit — optimization,
networks, stochastic models, queues,
inventory, simulation and decision theory — from
the modelling idea up. It assumes comfort with
Eleven stages, from writing your first model to steering a stochastic system. We begin with deterministic optimization — linear, network and integer programming, the beating heart of OR — then turn to dynamic and stochastic models where time and chance enter, and finish with simulation and the mathematics of decisions and games.
Every OR study is the same three-step dance: build a model, solve it, and translate the answer back into a decision. We start by learning to write that model — decision variables, an objective, and the constraints that bind them.
A linear program maximises a linear objective over linear constraints. It is the single most useful model in OR — and, remarkably, its optimum always sits at a corner of the feasible region, which is what makes the simplex method possible. Its dual hands you the shadow prices that tell a manager what each scarce resource is really worth.
A huge fraction of real OR problems live on a network — shipping goods, matching workers to jobs, routing data, scheduling a project. Many have such special structure that they solve far faster than a general LP, often with integer answers for free.
When decisions are yes/no or must come in whole units — build the factory or not, which trucks to dispatch — the variables turn integer, and the smooth world of LP shatters into a combinatorial explosion. This stage is the art of taming it.
Not every objective is a straight line. When costs curve — economies of scale, risk penalties, smooth trade-offs — we need the calculus of optimization: convexity (which guarantees a global optimum), the KKT conditions, and the iterative methods that climb to the top. This stage draws entirely on the existing optimization branch.
Many decisions are really a sequence of decisions unfolding over stages — and dynamic programming conquers them with one deceptively simple idea: an optimal plan is built from optimal sub-plans. Solve the little pieces once, remember them, and a problem that looked exponential collapses.
Now chance enters. A Markov chain models a system that hops between states with fixed probabilities; the Poisson process counts random arrivals in time; and a Markov decision process fuses randomness with control, becoming the backbone of both OR and reinforcement learning.
Every call centre, checkout, server farm and A&E department is a queue. Queueing theory predicts how long the line gets and how long you wait — and delivers the counter-intuitive truth that a system running near full capacity waits catastrophically longer than one with a little slack.
Hold too much stock and you tie up cash; hold too little and you lose the sale. Inventory theory finds the sweet spot — the classic economic order quantity, the one-shot newsvendor, and the reorder points that buffer against uncertain demand.
When a system is too tangled for a clean formula — as most real ones are — we simulate it: build a computational model, feed it random inputs, and watch what happens thousands of times. Done right, it estimates anything; done carelessly, it lies with great confidence.
Finally, the mathematics of choice. Decision analysis structures a choice under uncertainty and prices the value of information; game theory handles the harder case where the uncertainty is another rational player trying to beat you.
We begin where the field itself began — with the wartime idea that a hard decision is really a mathematical problem in disguise, waiting to be modelled and solved.