Mathematical Programming Formulations Based on McCarl and Spreen

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

Duality

Unbounded Solution

Infeasible Solution x1 x2 A B

Multiple Optima x1 x2 P1 P2 Isocline with highest objective

Degeneracy x1 x2 P1

Complementary Slackness derived from duality derived from duality

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

Multi-input, Multi-output X=Product Sale Y=Production Alternative Z=Input Purchase

Mixing / Blending F=Feed UL=Upper Limit LL=Lower Limit

Commodity Trade Y=Consumption (tons) X=Production (hectares) T=Trade (tons)

Sequencing X=Production Y=Sales t=Resource Endowment a, b, c, d, e, f=Resource requirements

Sequencing X=Planting Y=Harvesting Z=Selling

Storage X=Sale H=Hold

Lexicographic preferences

Weighted Preferences

Well behaved, Separable Function

Either-Or-Active constraints

Fixed Costs

Mutual exclusive products

Fixed Capacity

Warehouse V=Warehouse Z,Y,X=Shipments

Minimum Habitat Size h min 0 Area Population HAB 0 HAB 1

Minimum Habitat Size

Distinct Variable Values

Badly behaved non-linear functions

Non-linear Programming Specification often straightforward Specification often straightforward Solving more difficult Solving more difficult – Scale (manual or automatic) – Place lower and upper bounds on variables to avoid division by zero, other illegal operations, and numerical overflow – Use starting values to avoid poor local extremes

Equilibrium ‑ Unknown Life

Disequilibrium – Known Life

Disequilibrium – Unknown Life