Operations Research

Quantitative decision-making for operations. Learn transportation and assignment problems, network optimisation (critical path), scheduling, queueing models (M/M/1: utilisation, queue length, waiting time), inventory models (EOQ), and decision theory (expected value, decision trees).

Factories, hospitals, data centres, supply chains and maintenance departments use OR models to reduce waiting, cost and inventory while improving service.

An analogy

Operations research is like planning the busiest day of a restaurant: how many cooks, how much stock, which orders first — using numbers instead of guesswork.

Transportation and assignment

Ship goods from supplies to demands at minimum cost (special LP); assign n workers to n jobs at minimum cost (Hungarian algorithm).

Queueing

Waiting grows explosively as utilisation → 1: at ρ = 0.9 the average queue is 8.1; at 0.99, 98. Little's law relates queue length and waiting time for any stable system.

Inventory

EOQ balances ordering and holding costs; reorder point = demand during lead time + safety stock. Safety stock covers demand variability.

Scheduling and decisions

Critical path method (CPM/PERT) finds the project duration and critical tasks; job-shop scheduling rules (SPT, EDD); decision trees and expected value (or utility) under uncertainty; simulation for complex systems.

Formulas

• M/M/1 queue: ρ = λ/μ; L = ρ/(1 − ρ); W = 1/(μ − λ); Wq = ρ/(μ − λ); Lq = ρ²/(1 − ρ)
• Little's law: L = λW
• EOQ: Q* = √(2DS/H)
• Expected value: EV = Σ pᵢ vᵢ

Worked example: Maintenance workshop queue

Given: Breakdowns λ = 8/h, repair rate μ = 10/h (M/M/1)

Find: ρ, L, W, Wq.

• ρ = 0.8
• L = 4 machines, W = 1/(10 − 8) = 0.5 h
• Wq = 0.4 h, Lq = 3.2

Answer: Average 4 machines down, 30 min each

Worked example: EOQ

Given: D = 12 000 units/yr, S = $50/order, H = $2.40/unit·yr

Find: Q*.

• Q* = √(2 × 12 000 × 50/2.4) = 707

Answer: ≈ 707 units per order (≈ 17 orders/yr)

Common mistakes

• Wrong: Planning for 100 % utilisation. — Right: Queues become infinite; plan spare capacity.
• Wrong: Using averages when variability dominates. — Right: Include variability (safety stock, queue models).

Applications

• Service systems: Staffing call centres, repair crews.
• Supply chains: Transport and inventory optimisation.
• Project management: Critical path scheduling.

Key takeaways

• Transportation & assignment as special LPs.
• M/M/1: L = ρ/(1 − ρ); Little's law.
• EOQ = √(2DS/H).
• CPM, decision trees, simulation.

Going further

• M/M/c queues: Multiple servers (Erlang C).
• Discrete-event simulation: Complex operations modelling.
• Markov decision processes: Sequential decisions under uncertainty.