INEN 420 Final Review

Transcription

1 INEN 420 Final Review Office Hours: Mon, May :00-3:00 p.m. Tues, May :45-2:00 p.m. (Project Report/Critiques due on Thurs, May 5 by 5:00 p.m.) Tuesday, April 28,

2 Final Exam: Wednesday, May 11, 8:00 10:00 a.m. Closed book, closed notes, closed neighbor! You will be required to answer 5 problems out of 6 Final exam will not be returned to you However, you can stop by to see how you did on the final exam on Thurs, May 12. 2

3 Modeling Linear Programs (Chapter 3) Basic linear algebra (preliquisite) LP Assumptions Proportionality, additivity, divisibility, certainty LP Formulation Decision variables Objective function: Max/Min Constraints Sign restrictions Lots of Applications: Transportation (airlines, railway, ), chemical processes, farming, diet, finance, inventory control, scheduling, blending, etc. 3

4 Basic Polyhedral Theory (Chapter 3) Convexity: Convex sets, convex functions Feasible region: Polyhedron Extreme points (corner points) Extreme directions Bounded, unbounded polyhedra 4

5 Solving Linear Programs (Chapter 4) The Graphical Method Two decision variables Min: Isocost line / Max: Isoprofit line Special cases: unique, alternative/multiple, unbounded, infeasible solutions 5

6 Solving Linear Programs (Chapter 4) The Simplex Method Converting an LP to standard form Basic and nonbasic variables Finding an initial bfs Adjacent bfs Entering variable, ratio test, pivoting, etc Special cases: unique, unbounded, alternative/multiple solutions Convergence, degeneracy The Big-M Simplex Method No readily available initial bfs The Two-Phase Simplex Method No readily available initial bfs 6

7 Sensitivity Analysis and Duality (Chapter 6) Two of the most important topics in Linear Programming (How changes in the LP parameters affect the current optimal solution) Important Formulas Optimality condition Feasibility Condition Changing Parameters of an LP Effect on current basis Objective function coefficients (nonbasic, basic vars) RHS Adding a new activity 7

8 Sensitivity Analysis and Duality (Chapter 6) Finding the dual of an LP Normal max LP, normal min LP Basic Duality Theory Weak duality Strong duality Economic Interpretation: dual/shadow prices Complementary Slackness (CS) binding, nonbinding constraints (primal/dual problems) Using CS to compute an optimal solution to the primal/dual given the optimal solution to the other Dual Simplex Method (not discussed) Taught in INEN 622 Linear Programming 8

9 Special Problems (Chapter 7) Transportation Problems Balancing Northwest corner method (provides an initial bfs) Transportation simplex method Transshipment Problems Convert to transportation problem Apply the transportation simplex method Assignment problems Balancing The Hungarian method 9

10 Network Models (Chapter 8) Basic Definitions Graph/network Shortest Path Problems Dijkstra s Algorithm Minimum Spanning Trees MST algorithm (Greedy algorithm) Maximum Flow Problems The Ford-Fulkerson method Add costs on the arcs: Minimum Cost Network Flow Problem Network Simplex Method 10

11 Other Topics to Learn Project Management: CPM, PERT (Chapter 8) Integer Programming (Chapter 9 INEN 668) Linear Programming (Chapter 10 INEN 622) Game Theory (Chapter 11) Nonlinear Programming (Chapter 12 INEN 623) Dynamic Programming (Chapter 13 INEN 623) Heuristic Techniques (Chapter 14 INEN 689) and many more! 11

12 Stochastic Programming The Certainty Assumption No Longer Holds! My Research Interests Operations Research I INEN 420 Linear Programming INEN 622 Integer Programming INEN 668 Probability and Statistics STAT 610 Large-Scale Optimization Stochastic Programming Stochastic Integer Programming Programming Skills C/C++ Application Area of Interest Software Engineering Concepts Probability and Statistics Linear Programming INEN Large-Scale Stochastic Optimization Coming this Fall 2005! 12