Probability · Optimization · Decisions · Games

Mathematics 1 for
Risk Management

Welcome to Mathematics 1 for Risk Management. On this website, you will find the course notes, together with interactive figures, for each 3-hour session. This 12-hour course develops the mathematical tools needed for supply-chain risk analysis. The course focuses on the mathematics, but every concept is attached to a supply-chain application. We will first introduce probability, expectation, cumulative distribution functions (CDFs), variance and quantiles. Then, we will review unconstrained and constrained optimization, which relies on the derivatives of functions. In the third session, we combine the concepts from the first two sessions to study choice under uncertainty, and we introduce the loss function. The last session applies game theory to supply-chain contracts.

Observed weekly demand for sunglasses

Weekly demand for sunglasses at a retailer, 52 observed weeks (Snyder & Shen companion data). When you press play, the weeks are added to the histogram one at a time, in randomized order.

number of weeks in each demand range running mean

This histogram summarizes the observed distribution of weekly demand. In Session 1, we will use these data to introduce probability distributions and measures of uncertainty. In Session 3, we will use a demand model to choose an order quantity. In Session 4, we will study how contracts affect that choice and how risk is shared between a retailer and a supplier.

The four sessions

Session 1 · 3h

Probability

Probability, expectation, CDFs, variance and quantiles, used to describe uncertain demand.

Demand uncertaintyPoisson arrivalsService levels & VaR
Session 2 · 3h

Optimization, or finding the best choice/decision

Unconstrained and constrained optimization, and the role of convexity in finding a minimum.

Cost minimizationEOQ modelOptimal cycle time
Session 3 · 3h

Choice Under Uncertainty

Loss functions and expected costs, applied to the newsvendor model and to inventory decisions.

Newsvendor model(r, Q) & safety stockYield uncertainty
Session 4 · 3h

Game Theory

Game theory and supply-chain contracts between a retailer and a supplier who each maximize their own profit.

Wholesale priceBuybackRevenue sharing

How this site works

Each session combines four elements:

An application: a concrete supply-chain situation. An interactive figure: use the controls to explore how the model behaves. The mathematics: definitions, results, and step-by-step derivations. A check: a short question to test your understanding, with an explanation after you answer.

Advice

Use the sliders to connect each formula to its meaning. For example, observe how the optimum changes when holding costs rise, or how safety stock increases rapidly as the service level approaches 100%. Then practise explaining the result in words and calculating it with numbers.

References & data

The course follows Snyder & Shen, Fundamentals of Supply Chain Theory (2e, Wiley, 2019) and Nahmias & Olsen, Production and Operations Analysis (Waveland, 2015). All datasets used in the figures are the official Snyder & Shen companion data. They contain demand series for dog food, novels, oxygen, batteries and sunglasses.