1. Welcome to 620-261 Introduction to Operations Research

2. 620-261: Introduction to Operations Research
Lecturer: Peter Taylor
Heads Office
Richard Berry Building
Tel: -
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

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 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
: Decision Making
: Operations Research Methods and Algorithms
: 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

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. 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. 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. f(x)
Global max
Global min X 

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

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

36. Tip But do not be dogmatic about it !!!!!
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. 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.

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. The scheme should tell us what to do at each “trial”, i. e
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. 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. 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. 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

43. 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)

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

45. 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)

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

47. Dynamics = ????
We have to specify the dynamics of 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

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

49. Complete Formulation
(Erase N)

50. Complete Formulation

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. 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?

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 2x1018 possible tours !!!!!!!!!!!!!!!

54. The Curse of Dimensionality!
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!