1. Mathematical Programming Formulations Based on McCarl and Spreen

2. 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

3. Important LP Equations

4. Important LP Derivatives

5. Duality

6. Duality

7. Unbounded Solution

8. Infeasible Solution x1 x2 A B

9. Multiple Optima x1 x2 P1 P2 Isocline with highest objective

10. Degeneracy x1 x2 P1

11. Complementary Slackness derived from duality derived from duality

12. 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)

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

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

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

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

17. Sequencing X=Planting Y=Harvesting Z=Selling

18. Storage X=Sale H=Hold

19. Lexicographic preferences

20. Weighted Preferences

21. Well behaved, Separable Function

24. Either-Or-Active constraints

25. Fixed Costs

26. Mutual exclusive products

27. Fixed Capacity

28. Warehouse V=Warehouse Z,Y,X=Shipments

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

30. Minimum Habitat Size

31. Distinct Variable Values

32. Badly behaved non-linear functions

34. 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

35. Equilibrium ‑ Unknown Life

36. Disequilibrium – Known Life

37. Disequilibrium – Unknown Life