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Welcome to 620-261 Introduction to Operations Research

Welcome to 620-261 Introduction to Operations Research. 620-261: Introduction to Operations Research. Lecturer: Peter Taylor Heads Office Richard Berry Building Tel: 8344 7887 E-mail: - p.taylor@ms.unimelb.edu.au Course due to: Moshe Sniedovich. Schedule. Lectures:

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Welcome to 620-261 Introduction to Operations Research

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  1. Welcome to620-261Introduction toOperations Research

  2. 620-261: Introduction to Operations Research • Lecturer: Peter Taylor • Heads Office • Richard Berry Building • Tel: 8344 7887 • E-mail: - p.taylor@ms.unimelb.edu.au • Course due to: Moshe Sniedovich

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

  4. Office Hours • Monday 2-3 PM • Wednesday 2-3PM • Friday 2-3PM • These may have to vary sometimes – see my assistant Lisa Mifsud

  5. Assessment • Assignments: 10% • Final Exam: 90%

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

  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. Lecture Notes • On Sale (Book Room) • If out-of-print, let me know

  9. Thou Shall Not

  10. Thou Shall Not

  11. Student Representative(SSLC) • Pizza!!!!! • Two meetings • Questionnaire

  12. Web Site http://www.ms.unimelb.edu.au/~dmk/620-261

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

  14. Computer Literacy • Applied Mathematics is computational. • I don’t expect any specific knowledge, but I do expect an open attitude to things computational.

  15. Perspective Universe Applied maths 620-261 OR

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

  17. Basic Characteristics • Applies scientific methods • Adopts a systems approach • Utilises a team concept • Relies on computer technologies

  18. OR Stream • 620-261: Introduction to Operations Research • 620-262: Decision Making • 620-361: Operations Research Methods and Algorithms • 620-362: Applied Operations Research • Probability and Statistics are useful other subjects to study

  19. and more • Honours • MSc • PhD

  20. Jobs • There is a shortage of people with OR skills • Graduates with these skills get good jobs

  21. Reading ..... • Appendix A • Appendix B • Appendix E • Chapters 1,2,3,4 • Web

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

  23. In Practice Formulation Monitoring Realization Modelling Implementation Solution Analysis

  24. Important Comment In 620-261: • Formulation and Modelling • Analysis and Solution

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

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

  27. Example

  28. We let* denote the set of optimal decisions associated with the optimization problem. That is * denotes the subset of  whose elements are an 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.

  29. 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

  30. Classification of Optimal Solutions Consider the 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.

  31. f(x) Global max Global min X 

  32. 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.

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

  34. Thus ......... Modelling = • opt = ? • f(x) = ? • Constraints = ?

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

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

  37. 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

  38. 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:

  39. 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 decideswhere the false coin will be as we implement the weighing scheme ?

  40. 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!

  41. 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 (nL,nR,no) nL nR no

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

  43. Thus, our objective function is f(x1,x2,...,xn):= n and opt=min. To complete the formulation of the problem we have to determine .

  44. Constraints Let sj := Number of coins left for inspection after the j-th weighing (j = 0,1,2,...,n) Then clearly, s0 := N (All coins are yet to be inspected) sn := 1 (Only false coin is left for inspection)

  45. xj {0,1,2,...,[sj-1/2)]} where [z]:= Integer part of z.

  46. Dynamics = ???? • We have to specify the dynamicsof the process: how the {sj} are related to the {xj}. • This is not difficult because we assume that Nature Plays Against Us: sj = max {xj , sj-1-2xj} xj xj sj-1-2xj

  47. (j-1) weighing: sj-1 coins left xj xj j-th weighing: sj-1-2xj sj = max {xj , sj-1-2xj}

  48. Complete Formulation (Erase N)

  49. Complete Formulation

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