Quick Revision
All seven units compressed to what you must be able to reproduce from memory. Read top to bottom in about twenty minutes.
Unit 1 — Linear Programming
Allocate limited resources among competing activities to optimise one linear objective. Four blocks, every time: decision variables → objective function → constraints → non-negativity.
Proportionality · Additivity · Divisibility (fails ⟹ ILP) · Certainty (tested by sensitivity analysis).
feasible regionbasic feasible solutioncorner pointslacksurplusartificial variable
Equations → two points per line → shade feasible region → find corner points → evaluate Z at each in a table → pick best. Solve intersections simultaneously, never read them off the graph.
Entering = most positive Cⱼ − Zⱼ. Leaving = smallest non-negative replacement ratio. New pivot row = old ÷ pivot element; every other row = old − (its pivot-column entry × new pivot row). Max stops at all Cⱼ − Zⱼ ≤ 0.
| Case | Signal |
|---|---|
| Multiple optima | Non-basic variable with Cⱼ − Zⱼ = 0 |
| Unbounded | No positive replacement ratio |
| Infeasible | Artificial variable stays basic at non-zero value |
| Degenerate | Basic variable with Quantity = 0 (tie in ratios) |
Max ↔ Min. m constraints → m dual variables; n variables → n dual constraints. RHS ↔ objective coefficients. Transpose the matrix. Max primal ⟹ dual constraints ≥.
Equality splits into two dual variables (+Y₃ − Y₄); a wrong-direction inequality is multiplied by −1, flipping every sign including the RHS.
ΔZ = shadow price × ΔRHS, valid inside the allowable range. Binding = slack 0 and shadow price ≠ 0. Reduced cost 0 ⟹ variable is basic. A coefficient inside its range keeps the quantities but changes Z.
shadow pricereduced costallowable rangebinding constraintpost-optimality
Unit 2 — Integer Linear Programming
LP plus an integrality restriction — because you cannot make 2.5 cars or select 0.4 of a project. It is the divisibility assumption of Unit 1 failing.
Pure (all integer) · Mixed (some integer) · Binary (0–1 decisions).
Never omit the closing line — Xᵢ are binary (0,1) or … and are integers. It is a full mark.
pure ILPmixed ILPbinaryfixed chargecutting stocklinking constraint
Unit 3 — Transportation & Assignment
Supply and demand given ⟹ transportation. One-to-one matching of rows to columns ⟹ assignment.
Transportation — supply ≤, demand =. Assignment — rows and columns both = 1.
Profit = Selling price(j) − Plant cost(i) − Transport cost(i,j), objective becomes Max.
balanceddummy rowcost matrixidle time matrixm + nm × n
Unit 3's solution algorithms — North-West Corner Rule, Least Cost Method, VAM, MODI, and the Hungarian Method — are named in the session plan but are not covered anywhere in your notes, and neither paper asked for them. See the Unit 3 scope note.
Unit 4 — Theory of Games
Two-person zero-sum: one number per cell = the row player's gain. Rows maximise, columns minimise.
Maximin (max of row minima) = Minimax (min of column maxima) ⟹ saddle point, V = that value; V = 0 ⟹ fair. Dominance: drop smaller rows, bigger columns. 2 × 2 mixed: p = (d − c)/D, V = (ad − bc)/D, D = (a + d) − (b + c).
Q3A both papers: formulate the 2 × 2 matrix from sentences. Unit 4 · Assignment 7
Unit 5 — Queuing Theory
(M/M/1):(∞/FCFS). Convert to per hour: μ = 60 ÷ service minutes. ρ = λ/μ must be < 1.
P₀ = 1 − ρ · Pₙ = (1 − ρ)ρⁿ · P(n ≥ k) = ρᵏ · Ls = λ/(μ − λ) · Lq = λ²/[μ(μ − λ)] · Ws = 1/(μ − λ) · Wq = λ/[μ(μ − λ)]. Check Ls − Lq = ρ.
Q5A both papers: all parameters, then the effect of a higher λ (waits rise much faster than λ). Unit 5 · Assignment 5
Unit 6 — Simulation
Cumulate → RN ranges from 00 (cum × 100 − 1) → convert random numbers → fill the table → average.
Lorry: carry unshipped units forward. Queue: start = MAX(arrival, previous end). Profit: price × sold + salvage × unsold − cost × stock − goodwill × short.
Q2 (10 marks) both papers — the safest big question. Unit 6 · Assignment 9
Unit 7 — Decision Theory & AHP
No probabilities ⟹ uncertainty criteria. Probabilities ⟹ EMV. Stages or a survey ⟹ decision tree.
Laplace · Maximax · Maximin · Savage regret (column-wise) · Hurwicz α·max + (1 − α)·min. EMV, EPPI, EVPI = EPPI − best EMV = least EOL. Trees: roll back. Survey worth it only if EVSI > its cost. AHP: reciprocals → column-normalise → row-average.
Q1C (AHP), Q1D (criteria), Q6A (tree), Q6B (Hurwicz) — 15 marks a paper. Unit 7 · Assignment 8
The last hour before the exam
- Rehearse the duality conversion once with an equality constraint. It is Q1A in every paper.
- Say the counting block out loud: variables multiply, constraints add.
- Re-read the special-case identification table — especially degenerate vs multiple optima.
- Re-read the sensitivity four-step template: name the rule, quote the number, do the arithmetic, state the meaning.
- Remind yourself: "Formulate" means stop at the model.
- Write the RN-range rule and the queue-simulation columns once from memory — Q2 is 10 marks.
- Say the five decision criteria and EVPI = EPPI − best EMV out loud.
Then go in. Q1 is compulsory; pick the four of Q2–Q7 you have practised most.