UMA035-Optimization Techniques (Even semester 2022-23)

Course  Objectives:  The main objective of the course is to formulate mathematical models and to understand solution methods for real life optimal decision problems. The emphasis will be on basic study of linear and nonlinear programming problems, Integer programming problem, Transportation problem, Two person zero sum games with economic applications and project management techniques using CPM.

Syllabus for the course UMA035: 

Book to be followed:

Text Books:

1. Chandra, S., Jayadeva, Mehra, A., Numerical Optimization and Applications, Narosa Publishing House, (2013).

 2. Taha H.A., Operations Research-An Introduction, PHI (2007). 

Recommended Books:

1. Pant J. C., Introduction to optimization: Operations Research, Jain Brothers (2004) 

2. Bazaarra Mokhtar S., Jarvis John J. and Shirali Hanif D., Linear Programming and Network flows, John Wiley and Sons (1990) 

3. Swarup, K., Gupta, P. K., Mammohan, Operations Research, Sultan Chand & Sons, (2010). 

4. Kasana, H.S., and Kumar, K.D., Introductory Operations research, Springer publication, (2004) 

5. Ravindran, D. T. Phillips and James J. Solberg: Operations Research- Principles and Practice, John Wiley & Sons, Second edn. (2005). 

Tentative Lecture Plan:

Lab Material: 

Reference Books:

• Getting Started with MATLAB by Rudra Pratap

1. Experiment 1 pdf Sheet

2. Experiment 2 pdf Sheet

3. Experiment 3 pdf Sheet

4. Experiment 4 pdf Sheet

5. Experiment 5 pdf Sheet

6. Experiment 6 pdf Sheet

7. Experiment 7 pdf Sheet

8. Experiment 8 pdf Sheet

9. Experiment 9 pdf Sheet

Practice sheets:

1. Practice sheet 1:

2.  Practice sheet 2:

3. Practice sheet 3:

4. Practice sheet 4:

5. Practice sheet 5:

6. Practice sheet 6:

Lecture Notes: 

• Lecture Slides_Unit-I_(Scope of Operations Research ):

1. (Lecture-1) Introduction to linear and non-linear programming (Prerequisites).

2. (Lecture-2) Formulation of LPP models.

• Lecture Slides_Unit-II_(Linear Programming):

3. (Lecture-3) Geometry of LP and Fundamental theorem of LPP (with proof)

4. (Lecture-4) General and Standard form of LPP(with restricted and unrestricted sign of decision variables)

5. (Lecture-5) Algebraic or Analytical method to solve an LPP]

6. (Lecture-6) Graphical method to solve an LPP and exceptional cases

7. (Lecture-7) Simplex method of LPP with illustrations

8. (Lecture-8) Simplex method (How to find missing entries) with illustrations

9. (Lecture-9) Two phase method of LPP with illustrations

10. (Lecture-10) Big-M method to solve the LPP with illustrations

11. (Lecture-11) Exceptional cases of simplex method (Non-existing feasible solution, unbounded solution, alternative optimal, degenerate solution etc. with illustrations

12. (Lecture-12) Formulation of LPP from optimal table & construction of simplex table with arbitrary B. F. S. with illustrations

13. (Lecture-13) Duality theory with illustrations

14. (Lecture-14) Complementary slackness conditions of an LPP-Duality Theory with illustrations

15. (Lecture-15) Sensitivity Analysis or Post optimally with illustrations (Change of cost vector, Change of RHS Vector constraints)

16. (Lecture-16) Sensitivity Analysis or Post optimally with illustrations (Addition of variables and Addition of constraints)

• Lecture Slides_Unit-III_(Integer Programming):

17. (Lecture-17) Gomory's cutting plane method for integer programming problems

18. (Lecture-18) Branch and bound method with illustrations

• Lecture Slides_Unit-IV_(Network Models):

19. (Lecture-19) Introduction and Formulation of Transportation Problems, Initial BFS of Transportation Problems (NWCR, LCM and VAM) with illustrations

20. (Lecture-20) Optimal solution via MODI method and Alternate optimal solution with illustrations

21. (Lecture-21) Special cases of transportation problems (Unbalanced, Maximization, Prohibited area, Degeneracy in T.P.) with illustrations

22. (Lecture-22) Optimal solution of Assignment problem by Hungarian method and Special cases with illustrations

23. (Lecture-23) Construction of networks with illustrations

24. (Lecture-24) Critical Path Method (CPM) with illustrations

25. (Lecture-25) Optimal scheduling (crashing) by FF limit method with illustrations

• Lecture Slides_Unit-VI_(Nonlinear Programming):

• Nonlinear Programming: Constrained Optimization (Unit –VI):

26. (Lecture-26) Nonlinear Programming: Quadratic form-Hessian matrix-Stationary point and Convex functions with illustrations

27. (Lecture-27) Nonlinear programming problems with equality constraints (Lagrange’s multiplier method) with illustrations

28. (Lecture-28) Non-linear programming problems with inequality constraints (KKT conditions) with illustrations

• Nonlinear Programming: Unconstrained Optimization (Unit –VI):

29. (Lecture-29) Unimodal functions, Fibonacci search method, Steepest Descent method with illustrations

• Lecture Slides_Unit-V_(Multi-objective Linear Programming):

30. (Lecture-30) Introduction to Multi-objective linear programming, efficient solution, efficient frontier with illustrations