Linear Program MAX C B X B + C NB X NB s.t. BX B + A NB X NB = b X B, X NB ≥ 0

Important LP Equations

Important LP Derivatives

Duality

Unbounded Solution

Infeasible Solution

Multiple Optima

Degeneracy

Complementary Slackness derived from duality

Reduced Cost Negative derivative of objective function with respect to a variable At optimality: –Zero for all basic variables –Non-negative for all non-basic variables (max) –Non-positive for all non-basic variables (max)

Multi-input, Multi-output

Mixing / Blending

Spatial Equilibrium (GAMS Ex.)

Sequencing

Storage

Lexicographic preferences

Weighted Preferences

Well behaved, Separable Function

Disequilibrium – Known Life

Disequilibrium – Unknown Life

Equilibrium ‑ Unknown Life

Fixed Costs

Fixed Capacity

Minimum Habitat Size

Warehouse

Mutual exclusive products

Either-Or-Active constraints

Distinct Variable Values

Badly behaved non-linear functions

Non-linear Programming Specification often straightforward Solving more difficult –scaling (manual vs. computer) –lower bounds to avoid division by zero and other illegal operations –local versus global extremes