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1 Welcome to 620-261 Introduction to Operations Research.

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Presentation on theme: "1 Welcome to 620-261 Introduction to Operations Research."— Presentation transcript:

1 1 Welcome to Introduction to Operations Research

2 : Introduction to Operations Research  Lecturer: Peter Taylor  Heads Office  Richard Berry Building  Tel:  -  Course due to: Moshe Sniedovich

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 2-3PM  Friday 2-3PM  These may have to vary sometimes – see my assistant Lisa Mifsud

5 5 Assessment  Assignments: 10%  Final Exam: 90%

6 6 Group Projects  You 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)  If out-of-print, let me know

9 9 Thou Shall Not

10 10 Thou Shall Not

11 11 Student Representative (SSLC)  Pizza!!!!!  Two meetings  Questionnaire

12 12 Web Site

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

14 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 15 Perspective Universe Applied maths OR

16 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 17 Basic Characteristics  Applies scientific methods  Adopts a systems approach  Utilises a team concept  Relies on computer technologies

18 18 OR Stream  : Introduction to Operations Research  : Decision Making  : Operations Research Methods and Algorithms  : Applied Operations Research  Probability and Statistics are useful other subjects to study

19 19 and more  Honours  MSc  PhD

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

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

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

23 23 In Practice Solution Formulation Realization Modelling Analysis Implementation Monitoring

24 24 Important Comment In :  Formulation and Modelling  Analysis and Solution

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

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

27 27 Example

28 28

29 29 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.

30 30 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

31 31 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.

32 32 f(x)  Global max Global min X

33 33 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.

34 34 Suggestion Try to think about optimization problems in terms of the format: Z*:= opt f(x) s.t constraints

35 35 Thus Modelling =  opt = ?  f(x) = ?  Constraints = ?

36 36 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 !!!!!

37 37 Example 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.

38 38 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

39 39  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:

40 40 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 ?

41 41 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!

42 42 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

43 43 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)

44 44 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 .

45 45 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)

46 46 x j  {0,1,2,...,[s j-1 /2)]} where [z]:= Integer part of z.

47 47 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

48 48 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 }

49 49 Complete Formulation (Erase N)

50 50 Complete Formulation

51 51 Examples of OR Problems  Example Towers of Hanoi  Task: Move the discs from left to right  Rules: – One disc at a time – No large disc on a small one

52 52  Example 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?

53 53 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 !!!!!!!!!!!!!!!

54 54  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!


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