Government Operational Research Service (GORS) Assessment Core Operational Research Techniques Syllabus

State the PMF, mean and variance of a Poisson distribution.

$P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}$ for $k = 0,1,2,\dots$, with mean $= \lambda$ and variance $= \lambda$. It models the number of rare independent events in a fixed interval.

What is the memoryless property and which distributions have it?

$P(X > s + t \mid X > s) = P(X > t)$ — the future is independent of elapsed time. The exponential distribution (continuous) and geometric distribution (discrete) are the only ones with this property.

State the Central Limit Theorem (CLT).

For i.i.d. random variables with mean $\mu$ and finite variance $\sigma^{2}$, the sample mean's distribution approaches Normal as $n \to \infty$: $\bar{X} \approx N\!\left(\mu, \frac{\sigma^{2}}{n}\right)$, regardless of the original distribution's shape.

What is a confidence interval, and how is a 95% CI for a mean (known $\sigma$) formed?

A range that would contain the true parameter in 95% of repeated samples. For a mean: $\bar{x} \pm z_{0.025}\,\frac{\sigma}{\sqrt{n}}$, i.e. $\bar{x} \pm 1.96\,\frac{\sigma}{\sqrt{n}}$.

Define Type I and Type II errors in hypothesis testing.

Type I error: rejecting a true null hypothesis (false positive), probability $\alpha$. Type II error: failing to reject a false null (false negative), probability $\beta$. Power $= 1 - \beta$ is the chance of correctly detecting a true effect.

What is a p-value, and how is it used in a hypothesis test?

The probability, assuming $H_0$ is true, of observing a test statistic at least as extreme as the one obtained. Reject $H_0$ if $p \leq \alpha$ (the significance level). It is NOT the probability that $H_0$ is true.

When do you use a t-test versus a z-test for a population mean?

Use a z-test when the population variance $\sigma^{2}$ is known (or $n$ is large). Use a t-test when $\sigma$ is unknown and estimated from a small sample; the t-distribution has heavier tails and depends on degrees of freedom $n-1$.

State the simple linear regression model and the least-squares slope.

Model: $y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$. The least-squares slope is $\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^{2}} = \frac{\operatorname{Cov}(x,y)}{\operatorname{Var}(x)}$, with intercept $\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}$.

What does the coefficient of determination $R^{2}$ measure?

The proportion of total variance in the response explained by the regression: $R^{2} = 1 - \frac{SS_{res}}{SS_{tot}}$, ranging from 0 to 1. In simple regression $R^{2} = r^{2}$, the square of the correlation coefficient.

What does Pearson's correlation coefficient $r$ measure, and its range?

The strength and direction of a linear association between two variables, $r = \frac{\operatorname{Cov}(x,y)}{\sigma_x \sigma_y}$, with $-1 \leq r \leq 1$. Note: correlation does not imply causation, and $r$ only captures linear relationships.

List the key assumptions of ordinary least squares (OLS) linear regression.

Linearity, independence of errors, homoscedasticity (constant error variance), normally distributed errors, and no perfect multicollinearity among predictors (the 'LINE' assumptions plus full rank).

Name the four classical components of a time series.

Trend (long-term direction), seasonality (regular periodic fluctuations), cyclical variation (longer irregular waves, e.g. economic cycles), and irregular/random noise. Decomposition can be additive ($Y = T + S + C + I$) or multiplicative ($Y = T \times S \times C \times I$).

What is exponential smoothing and its updating formula?

A forecasting method that weights recent observations more heavily: $\hat{y}_{t+1} = \alpha y_t + (1-\alpha)\hat{y}_t$, where $0 < \alpha < 1$ is the smoothing constant. Higher $\alpha$ reacts faster to recent change.

What does the ARIMA$(p,d,q)$ model's notation mean?

$p$ = order of the autoregressive (AR) part, $d$ = degree of differencing to achieve stationarity (I), and $q$ = order of the moving-average (MA) part. It combines past values, differencing, and past forecast errors.

Define stationarity in a time series and why it matters.

A series is (weakly) stationary if its mean, variance and autocovariance are constant over time and do not depend on $t$. Many forecasting models (e.g. ARMA) assume stationarity; non-stationary series are often differenced to achieve it.

In Kendall's notation $A/B/c/K/N/D$ for a queue, what do the symbols mean?

$A$ = arrival process distribution, $B$ = service time distribution, $c$ = number of servers, $K$ = system capacity, $N$ = population size, $D$ = queue discipline. M denotes Markovian (exponential), e.g. M/M/1 is Poisson arrivals, exponential service, one server.

For an M/M/1 queue with arrival rate $\lambda$ and service rate $\mu$, give utilisation and mean number in system.

Traffic intensity $\rho = \frac{\lambda}{\mu}$ (stable if $\rho < 1$). Mean number in system $L = \frac{\rho}{1-\rho} = \frac{\lambda}{\mu - \lambda}$.

State Little's Law and what it relates.

$L = \lambda W$: the average number of items in a stable system equals the average arrival rate times the average time an item spends in the system. It holds regardless of the arrival or service distribution.

What defines a (discrete-time) Markov chain?

A stochastic process where the next state depends only on the current state, not the history: $P(X_{n+1} = j \mid X_n = i, \dots) = P(X_{n+1} = j \mid X_n = i) = p_{ij}$. This is the Markov (memoryless) property, encoded in a transition matrix.

What is a stationary (steady-state) distribution of a Markov chain?

A probability vector $\vec{\pi}$ satisfying $\vec{\pi}P = \vec{\pi}$ with $\sum_i \pi_i = 1$. For an irreducible, aperiodic chain it is unique and gives the long-run proportion of time spent in each state.

How do you compute the $n$-step transition probabilities of a Markov chain?

By raising the one-step transition matrix to the $n$-th power: the $(i,j)$ entry of $P^{n}$ gives $P(X_n = j \mid X_0 = i)$. This follows from the Chapman–Kolmogorov equations.

Distinguish absorbing, transient, and recurrent states in a Markov chain.

Absorbing: once entered, never left ($p_{ii} = 1$). Recurrent: the chain returns with probability 1. Transient: there is positive probability of never returning. Absorbing chains are used to compute expected time to absorption.

What is Monte Carlo simulation and a core convergence property?

A method that estimates quantities by repeated random sampling — e.g. estimating $E[f(X)]$ by averaging $f$ over many random draws. The estimation error shrinks as $\frac{1}{\sqrt{n}}$ (standard error $\propto \frac{\sigma}{\sqrt{n}}$), independent of dimension.

How is inverse transform sampling used to generate random variates?

To sample from a distribution with CDF $F$, draw $U \sim \text{Uniform}(0,1)$ and set $X = F^{-1}(U)$. Then $X$ has the desired distribution, since $F$ maps uniformly distributed probabilities back to values.

What distinguishes Discrete Event Simulation (DES) from continuous simulation?

DES models a system as a sequence of discrete events that change state at specific instants (e.g. arrivals, service completions), advancing the clock event-by-event. Continuous simulation evolves state continuously over time (e.g. differential equations, system dynamics).

What are the key components of a discrete event simulation model?

Entities (objects flowing through), events (state changes), the simulation clock, a future event list, queues, resources/servers, and random number generators for stochastic inputs. The next-event time-advance mechanism jumps the clock to the next scheduled event.

What is a decision tree and how is it evaluated?

A diagram with decision nodes (squares, choices), chance nodes (circles, probabilistic outcomes), and end payoffs. It is evaluated by 'rolling back' (backward induction): take expected values at chance nodes and the best choice at decision nodes.

Define the Expected Value of Perfect Information (EVPI).

$\text{EVPI} = (\text{expected payoff with perfect information}) - (\text{expected payoff of the best action without information})$. It is the maximum a rational decision-maker should pay for information that fully removes uncertainty.

State the maximin, maximax, and minimax-regret decision criteria under uncertainty.

Maximin: choose the action with the best worst-case payoff (pessimistic). Maximax: choose the action with the best best-case payoff (optimistic). Minimax regret: minimise the maximum opportunity loss (regret) across states.

Distinguish 'decisions under risk' from 'decisions under uncertainty'.

Under risk, probabilities of outcomes are known, so you can use expected value / expected utility. Under uncertainty, probabilities are unknown, so criteria like maximin, maximax, Hurwicz, or minimax regret are applied instead.

What is Multi-Criteria Decision Analysis (MCDA) and the basic weighted-sum model?

A framework for evaluating options against multiple, often conflicting, criteria. The weighted-sum (linear additive) model scores each option as $S_j = \sum_i w_i \, s_{ij}$, where $w_i$ are criterion weights ($\sum_i w_i = 1$) and $s_{ij}$ are normalised scores.

What is the Analytic Hierarchy Process (AHP) in MCDA?

An MCDA method that structures a decision into a hierarchy (goal, criteria, alternatives) and derives weights from pairwise comparisons using a 1–9 scale. It computes priorities from the principal eigenvector and checks a consistency ratio.

What is cost-benefit analysis (CBA) and the role of Net Present Value?

CBA appraises options by valuing all costs and benefits in monetary terms over time. The decision rule uses Net Present Value: $NPV = \sum_{t=0}^{T} \frac{B_t - C_t}{(1+r)^{t}}$; accept projects with $NPV > 0$ (or the highest NPV).

What is the HM Treasury Green Book and its recommended discount rate?

The UK government's guidance on appraisal and evaluation of policies, programmes and projects. It recommends a standard Social Time Preference Rate (discount rate) of 3.5% for most appraisals (declining for very long horizons), and use of social (not just financial) value.

In the Green Book, what is optimism bias and how is it handled?

Optimism bias is the demonstrated tendency for appraisers to be over-optimistic about costs, benefits and timescales. The Green Book requires explicit, empirically based uplift adjustments to costs to counteract it.

What are the building blocks of system dynamics modelling?

Stocks (accumulations/levels), flows (rates that change stocks), feedback loops (reinforcing R and balancing B), and time delays. It models how nonlinear feedback and accumulation drive system behaviour over time.

Distinguish reinforcing and balancing feedback loops in system dynamics.

Reinforcing (positive) loops amplify change, producing exponential growth or decline (e.g. compound interest). Balancing (negative) loops counteract change, seeking equilibrium or a goal (e.g. a thermostat). Their interaction generates complex dynamics.

What is the Critical Path Method (CPM) and the critical path?

A project-scheduling technique that finds the longest sequence of dependent activities (the critical path), which determines the minimum project duration. Activities on it have zero float/slack; delaying any of them delays the whole project.

How is total float (slack) of an activity calculated in CPM?

Total float $= LS - ES = LF - EF$, where ES/EF are earliest start/finish and LS/LF are latest start/finish. It is the amount an activity can be delayed without delaying project completion; critical activities have zero float.

How does PERT estimate an activity's expected time and variance?

Using three estimates — optimistic $a$, most likely $m$, pessimistic $b$: expected time $t_e = \frac{a + 4m + b}{6}$ and variance $\sigma^{2} = \left(\frac{b - a}{6}\right)^{2}$. PERT treats durations probabilistically (Beta distribution).

State the Economic Order Quantity (EOQ) formula and what it minimises.

$EOQ = \sqrt{\dfrac{2DS}{H}}$, where $D$ = annual demand, $S$ = ordering cost per order, $H$ = holding cost per unit per year. It minimises the total of ordering plus holding costs, where these two costs are equal at the optimum.

What is the reorder point in inventory control (with safety stock)?

Reorder point $= d \cdot L + SS$, where $d$ is average demand per unit time, $L$ is lead time, and $SS$ is safety stock held to buffer against demand/lead-time variability. An order is placed when inventory falls to this level.

What is the bullwhip effect in supply chains?

The amplification of demand variability as orders move upstream from retailer to manufacturer: small fluctuations in consumer demand cause progressively larger swings in orders and inventory upstream, driven by batching, forecasting lags, and price fluctuations.

What is a Nash equilibrium in game theory?

A set of strategies (one per player) where no player can improve their payoff by unilaterally changing their own strategy, given the others' strategies. Each player's choice is a best response to the others'.

Explain the Prisoner's Dilemma and its key lesson.

Two players each choose cooperate or defect; mutual cooperation beats mutual defection, but defecting dominates individually. Both rationally defect, reaching a Pareto-inferior Nash equilibrium — illustrating how individual rationality can produce a collectively worse outcome.

Distinguish a dominant strategy from a dominated strategy.

A dominant strategy yields a player a payoff at least as high as any other strategy regardless of opponents' choices. A dominated strategy is always worse than some alternative and can be eliminated. Rational players play dominant strategies and avoid dominated ones.

What is the difference between a pure strategy and a mixed strategy?

A pure strategy deterministically selects one action. A mixed strategy assigns probabilities over actions, randomising the choice. Nash's theorem guarantees that every finite game has at least one equilibrium in (possibly mixed) strategies.

What is sensitivity analysis in an LP/optimisation context?

Examining how the optimal solution and objective value change as model inputs (objective coefficients, RHS values, constraints) vary. It identifies ranges over which the current basis/solution stays optimal and quantifies robustness to data uncertainty.

What is the difference between expected value and expected utility decision-making?

Expected value weights monetary outcomes by probability and assumes risk-neutrality. Expected utility applies a utility function $u(x)$ first (often concave for risk-aversion), so $E[u(X)]$ can rank a sure smaller payoff above a risky larger-EV gamble.

What is autocorrelation and why does it matter in time series and regression?

Autocorrelation is the correlation of a series with its own lagged values. In time series it reveals structure (used in ARIMA identification via ACF/PACF); in regression it violates the independence-of-errors assumption, biasing standard errors (tested via Durbin–Watson).