Geraint Palmer Optimisation using Linear Programming

OR Methods “Soft” methods Methods to help structure ill- structured problem situations Methods for more structured problems -parameters difficult to quantify Methods to calculate an attribute of a system Deterministic methods Stochastic methods Statistical methods Static Monte Carlo simulation methods Probabilistic methods Logic methods Methods to replicate or forecast system behaviour Deterministic replication methods Stochastic replication methods Complexity understanding methods Optimization methods Optimization of deterministic systems Optimization of stochastic systems Taxonomy of OR methods Williams, T (2008) Management Science in Practice, Wiley. p.101

Given a restricted budget, how much of drug A and drug B should a hospital purchase in order to maximise number of QALYs saved? A hospital can save 50 QALYs per tonne of drug A, and 60 QALYs per tonne of drug B. A tonne of A requires 4 tonnes of X and 5 tonnes of Y. A tonne of B requires 6 tonnes of X and 4 tonnes of Y. We can only afford 24 tonnes of X, and 20 tonnes of Y. Problem

Given a restricted budget, how much of drug A and drug B should a hospital purchase in order to maximise number of QALYs saved? A hospital can save 50 QALYs per tonne of drug A, and 60 QALYs per tonne of drug B. A tonne of A requires 4 tonnes of X and 5 tonnes of Y. A tonne of B requires 6 tonnes of X and 4 tonnes of Y. We can only afford 24 tonnes of X, and 20 tonnes of Y. Maximise: 50 A + 60 B Constraints: Resource X: 4 A + 6 B ≤ 24 Resource Y: 5 A + 4 B ≤ 20 Formulation

4 A + 6 B ≤ 24 5 A + 4 B ≤ 20 Solution

4 A + 6 B ≤ 24 5 A + 4 B ≤ A + 60 B Solution

4 A + 6 B ≤ 24 5 A + 4 B ≤ A + 60 B Solution

Objectives and constraints must be linear Multi-Dimensional – More variables! Lots of Constraints Computer does all the solving Mixed Integer Linear Programming – when solutions must be integers Sensitivity Analysis Features

Potential Uses Resource Allocation Operating Room Scheduling Nurse Rostering Workforce Planning Public Health Campaigns