Modeling and Optimization
This domain covers linear and integer programming, network models, decision analysis, queueing, and simulation.
Linear programming
Formulate decision variables, objective, constraints, and nonnegativity. With two variables, solve graphically: the optimum lies at a vertex, and iso-profit lines are parallel to the objective's slope. Simplex concepts: slack variables for ≤ constraints, surplus and artificial variables for ≥. A binding constraint has zero slack. The shadow price is the marginal value of one more unit of the right-hand side, valid only within the allowable range. Reduced cost is the rate at which the objective changes if a nonbasic variable is forced into the solution.
Integer and network models
If variables must be whole (people, machines, trucks), rounding an LP answer is not valid - use integer programming. Transportation and assignment models are LP special cases; unbalanced problems need a dummy row or column. Shortest path, minimum spanning tree, and maximum flow are distinct problems with distinct algorithms.
Decision analysis
Decision trees roll back from terminal nodes using expected monetary value. EVPI = EV with perfect information − EV without; EVSI is always ≤ EVPI.
Queueing
M/M/1 requires ρ = λ/μ < 1. Then Lq = ρ²/(1−ρ), L = ρ/(1−ρ), and W = L/λ. Reducing variability or adding a server cuts waiting far more than small utilization tweaks.
Traps
Maximizing when minimizing was asked; ignoring which constraint is binding; extrapolating a shadow price beyond its range; choosing on EMV alone when the decision maker is risk-averse.
Worked example
λ = 8/hr and μ = 10/hr gives ρ = 0.8, Lq = 0.64/0.2 = 3.2 customers waiting, L = 4 in the system, and W = 4/8 = 0.5 hour.