1. 1 Welcome to 620-261 Introduction to Operations Research

2. 2 620-261: Introduction to Operations Research  Lecturer:  Moshe Sniedovich  Room 150  Richard Berry Building  Tel: 8344 5559  E-mail: – m.sniedovich@ms.unimelb.edu.au

3. 3 Schedule  Lectures: Mon, Wed, Friday 3:15 PM  Tutorial: Check Notice Board and Web site

4. 4 Office Hours  Monday 2-3 -PM  Wednesday 11AM-12PM  Friday 2-3PM  Other times by appointment  Walk in......

5. 5 Assessment  Assignments (weekly): – 10%  Final Exam: – 90%  Class Performance: – on the fly

6. 6 Group Projects  Your are encouraged to study with friends, but you are expected to compose your own reports.

7. 7 Communication  You are expected to respond to questions asked (by the lecturer) during the lectures  Suggestions, comments, complaints: – Directly to lecturer – via Student Representative  Don’t wait till you are asked to complain!

8. 8 Lecture Notes  On Sale (Book Room)  Limited supply  If out-of-print, let us know

9. 9 Thou Shall Not

10. 10 Thou Shall Not

11. 11 Thou Shall Not

12. 12 Thou Shall Not

13. 13 Thou Shall Not

14. 14 Thou Shall Not

15. 15 Thou Shall Not

16. 16 Student Representative (SSLC)  Pizza!!!!!  Two meetings  Questionnaire

17. 17 Web Site http://www.ms.unimelb.edu.au/~moshe/620-261/

18. 18 Reference Material  Lecture Notes  Bibliography (10 copies of Winston in Maths library, reserved Shelves)  Hand-outs

19. 19 Computer Literacy

20. 20 Perspective Universe Applied maths OR 620-261

21. 21 What is OR ?  Controversial question!  Surf the WWW for answers  Roughly: –.... Applications of quantitative scientific methods to decisions making and support in business, industrial and military organisations, with an objective of improving the quality of managerial decisions.....

22. 22 Basic Characteristics  Applies Scientific methods  Adopts a Systems Approach  Utilises a team concept  Relies on computer technologies

23. 23 Computers in OR  Management Information Systems (MIS)  Database Management (DBM)  Decision Support Systems (DSS)  Expert Systems (ES)  Artificial Intelligence (AI)  Geographic Information Systems (GIS)

24. 24 OR Stream  620-261: Introduction to Operations Research  620-262: Decision Making  620-361: Operations Research Methods and Algorithms  620-362: Applied Operations Research

25. 25 and more  Honours  MSc  PhD  Related topics in statistics

26. 26 Jobs

27. 27 Reading.....  Appendix A  Appendix B  Appendix E  Chapters 1,2,3,4  Web

28. 28 The OR Problem Solving Schema Solution Formulation Realization Modelling Analysis Implementation Monitoring

29. 29 In Practice Solution Formulation Realization Modelling Analysis Implementation Monitoring

30. 30 Important Comment In 620-261:  Formulation and Modelling  Analysis and Solution

31. 31 Introducing....

32. 32 Chapter 2: Optimization Problems  General formulation f Objective function x Decision variable  Decision Space opt Optimality criterion z* Optimal return/cost

33. 33  Observe the distinction between f and f(x).  Note that f is assumed to be a real valued function on .

34. 34 Example

35. 35

36. 36 We let  * denote the set of optimal decisions associated with the optimization problem. That is  * denotes the subset of  whose elements are optimal solution to the optimization problem. Formally,  *:={x*: x* , f(x*)=opt {f(x): x   }}. By construction  * is a subset of , namely optimality entails feasibility.

37. 37 Remarks  The set of feasible solution, , is usually defined by a system of constraints.  Thus, an optimization problem has three ingredients: – Objective function – Constraints – Optimality Criterion

38. 38 Classification of Optimal Solutions Consider e case where opt=min. Then by definition: x*   * iff f(x*)  f(x)  x   If opt=max: x*   * iff f(x*) ≥ f(x)  x   Solutions of this type are called global optimal solutions.

39. 39 f(x)  Global max Global min X

40. 40 Question: How do we solve optimization problems of this type? Answer: There are no general purpose solution methods. The methods used are very much problem-dependent.

41. 41 Suggestion Try to think about optimization problems is terms of the format: Z*:= opt f(x) s.t.----------------------------- ----------------------------- constraints -----------------------------

42. 42 Thus......... Modelling =  opt = ?  f(x) = ?  Constraints = ?

43. 43 Tip You may find it useful to adopt the following approach:  Step 1: Identify and formulate the decision variables.  Step 2: Formulate the objective function and optimality criterion.  Step 3: Formulate the constraints. But do not be dogmatic about it !!!!!

44. 44 Example 2.4.2 False Coin Problem  N coins  N-1 have the same weight (“good”)  1 is heavier (“false”)  Find the best weighing scheme using a balance beam.

45. 45 Observations  It does not make sense to put a different number of coins on each side of the scale.  The result of any non-trivial weighing must fall into exactly one of the following cases: – False coin is on the left-hand side – False coin is on the right-hand side – False coin is not on the scale

46. 46  The scheme should tell us what to do at each “trial”, i.e. how many coins to place on each side of the scale, depending on how many coins are still to be inspected.  The term “Best” needs some clarification:

47. 47 Best = ???  “Best” = “fewest number of weighings” is not well defined because a priori we don’t know how many weighings will be needed by a given scheme.  This is so because we do not know where the false coin will be placed.  The bottom line: who decides where the false coin will be as we implement the weighing scheme ?

48. 48 We need help!!!  Many of the difficulties are nicely resolved if we assume that  Mother Nature Always Plays Against Us!  Of course, if you are an optimist you may prefer to assume that  Mother Nature Always Plays in Our Favour!

49. 49 Assumption Mother Nature Always Plays Against Us ! Observe that this assumption resolves the question of where the false coin will be. Nature will always select the largest of (n L,n R,n o ) nLnL nRnR nono

50. 50 Solution Let n := Number of weighings required to identify the false coin. x j := Number of coins placed on each side of the scale in the j-th weighing (j=1,2,3,...,n)

51. 51 Thus, our objective function is f(x 1,x 2,...,x n ):= n and opt=min. To complete the formulation of the problem we have to determine .

52. 52 Constraints Let s j := Number of coins left for inspection after the j-th weighing (j = 0,1,2,...,n) Then clearly, s 0 := N (All coins are yet to be inspected) s n := 1 (Only false coin is left for inspection)

53. 53 x j  {0,1,2,...,Int((s j-1 )/2)} where Int(z):= Integer part of z.

54. 54 Dynamics = ????  We have to specify the dynamics of the process: how the {s j } are related to the {x j }.  This is not difficult because we assume that Nature Plays Against Us: s j = max {x j, s j-1 -2x j } xjxj xjxj s j-1 -2x j

55. 55 xjxj xjxj s j-1 -2x j j-th weighing: (j-1) weighing: s j-1 coins left s j = max {x j, s j-1 -2x j }

56. 56 Complete Formulation (Erase N)

57. 57 Complete Formulation

58. 58 Comments  This problem is a good example of a situation where it is much easier to solve the problem than to formulate it mathematically.  Make sure that you fully understand and appreciate the difference between “solving” and “formulating” a problem.

59. 59 Examples of OR Problems  Example 2.4.6 Towers of Hanoi  Task: Move the discs from left to right  Rules: – One disc at a time – No large disc on a small one

60. 60  Example 2.4.4 Travelling Salesman Problem Visit N cities, starting the tour and terminating it in the home city such that: Each city (except the home city) is visited exactly once The tour is as short as possible. Question: What is the optimal tour?

61. 61 Remark: There are (N-1)! distinct tours. This means that for 11 cities there are 3,628,800 possible tours and for N=21 cities there are 2x10 18 possible tours !!!!!!!!!!!!!!!

62. 62  If we try to enumerate all the feasible tours for N=21 using a super fast computer capable of enumerating 1,000,000,000 tours per second, we will complete the enumeration of all the feasible tours in approximately 800 years.  This phenomenon is known as The Curse of Dimensionality!