part 1 overview, introduction, examples what is operations research? what is optimization what is...
TRANSCRIPT
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Part 1 Overview, introduction,
examples
• What is Operations Research?• What is Optimization• What is Sequential Decision
Making?• What is Dynamic Programming?• Examples, please!
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Perspective
Universe
AppliedMaths
OR
OptimizationSequential Decision
Making
620-113
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What is Operations Research?
• A branch of applied mathematics dealing with the Science of Decision Making
• Also known as Management Science• One of the streams at the Maths & Stats
Department• More information at www.ifors.org• Local chapter
www.asor.ms.unimelb.edu.au/melbourne
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What is optimization?• Extensive literature (books,
journals, encyclopedia)• Education and training programs• Jobs• Commercial software (eg. Excel)• Consulting services• Conferences and workshops• Meeting place: e-optimization.com
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Convention• Universal Convention for the description
of optimisation problems
opt Objective (dvs) dvs
Subject to: Constraints (dvs)
• Opt: either min or max.• Objective: an expression
(function) quantifying how good/bad our decisions are.
• dvs: decision variables• Constraints:
Restrictions on the decision variables.
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Help Desk
opt Objective (dvs) dvs
Subject to: Constraints (dvs)
TipUsually it is best to build
the model in the following order:– dvs (Decision variables)– Objective– Opt– Constraints But, ….. do not be dogmatic!
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Naïve ExampleEnglish Version
• Given:The value of the perimeter of a rectangle
• Task: Determine the width and height of the rectangle that will make its area as large as possible
Maths Version
maxw,h
A=w×h
s.t.
2(w+h) =p
Which one do you like better?
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Help Desk
maxw,h
A=w×h
s.t.
2(w+h) =p
Formally• Opt = max • dvs: w,h• Objective: w x h• Constraint:
2(w + h) = p
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Help Desk
h
w
Perimeter = 2(w+h)Area = w x h
maxw,h
A=w×h
s.t.
2(w+h) =p
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Beware of the distinction between
• Problem FormulationDescribing what the
problem is all about
• Problem SolvingConstructing a
solution to a given problem
• Most students under estimate the difficulties encountered at the formulation stage
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Naïve Example
English Version• Given:The value of the
perimeter of a rectangle
• Task: Determine the width and height of the rectangle that will make its area as large as possible
Maths Version
maxw,h
A=w×h
s.t.
2(w+h) =p
This is the (maths) model
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Observations
maxw,h
A=w×h
s.t.
2(w+h) =p
There are infinitely many feasible solutions: Example
2(w + h) = p = 6 w = 3 - h/2We can let h take
value in the range (0,3).
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Observations
• Not difficult to show that the optimal solution is
maxw,h
A=w×h
s.t.
2(w+h) =pw=h=p4
A=p2
16
I am asquare Prove this!
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Strong
Operations Research
Flavour
Warning
Parental Guidance is required
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Role of Models in Problem SolvingRealisation
InformalDescription
FormalModel Solution
WorkingSystem
Modelling
Analysis
Implementation
Monitoring
mathematical
^
Don’t panic!
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In Real-life
Realisation
InformalDescription
FormalModel Solution
WorkingSystem
Modelling
Analysis
Implementation
Monitoring
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We shall focus on
Realisation
InformalDescription
FormalModel Solution
WorkingSystem
Modelling
Analysis
Implementation
MonitoringGiven
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Beware of the distinction between
• Problem FormulationDescribing what the
problem is all about
• Problem SolvingConstructing a
solution to a given problem
• Most students under estimate the difficulties encountered at the formulation stage
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Example (Knapsack Problem)
English Version• Given: A container
and a collection of items having known values and volumes.
• Task: Determine what is the sub-collection of items of maximum total value that will fit into the container.
Maths Version
?????
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5 minutes
English Version• Given: A container
and a collection of items having known values and volumes.
• Task: Determine what is the sub-collection of items of maximum value that will fit into the container.
Maths Version
Help!
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What is a Sequential Decision Problem ?
• A problem (typically an optimization problem) representing a sequential decision making situation.
• Example: Knapsack problem.• Note the distinction between the problem
itself and the formulation given to it.
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What is a Dynamic Programming?
Short version• A paradigm for the
formulation, analysis and solution for sequential decision problems.
Answer
Long version• Wait and see …
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• Difficult to teach• Difficult to learn• Usually done “by example”
Important Facts about DP
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DP Approach to Problem Solving
• Step 1: Transform your problem into a family of related problems
• Step 2: Derive a functional equation relating the solutions to these problems to each other
• Step 3: Solve the functional equation• Step 4: Recover from the functional
equation a solution to your initial problem
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Trouble is,
This is easier said than done
The Art of DP
hence
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Most Famous Example(Shortest Path Problem)
• Given: a set of nodes (cities) and the lengths of the arcs connecting them
• Task: Find the shortest path from some given node to another given node.
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
1 2 1 2
1
2
10152010
1 2 1 2
110152010
121
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Step 1: Create a Family of Related Problems
• Suppose we are given say m cities and have to determine the shortest path from city s to city d.
• Let f(j) := shortest distance from city s
to city j, j=1,2,3,…,m. Note: we now have m problems.
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Step 2: Deriving a Functional Equation
• From the definition of f(j) it follows that f(j) = t(k,j) + f(k)
for some immediate predecessor k of j.• Hence,
f( j) =mink∈P ( j )
{t(k, j)+f(k)}
where t denotes the distance matrix and P(j) denotes the set of immediate predecessors of node j.
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Example
f(8) = mink∈P (8)
{t(k,8)+f(k)}
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
1 2 1 2
1
2
10152010
1 2 1 2
110152010
121
=mint(3,8)+f(3)
t(7,8)+f(7)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
=min15+f(3),2+f(7){ }
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Step 3: Solve the FE
• Can be– Very easy– Easy– Difficult– Very difficult– Impossible
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Example (naïve)1
13
D
E
A B
C
1
2
• f(A) = 0 (by definition?)• f(B) = t(A,B) + f(A) = 3 + 0 = 3• f(C) = t(B,C) + f(B) = 1 + 3 = 4• f(D) = t(B,D) + f(B) = 1 + 3 = 4• f(E) = min {t(C,E) + f(C),t(D,E) +
f(D)} = min {1+4,2+4} = 5
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Step 4: Recover a solution
• Once the functional equation is solved, it is usually simple to recover an optimal solution to the original problem. (Hint: go in …. reverse)
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Example (naïve)
1
1
3
D
E
A B
C
1
2
• Optimal solutions to the FE (going from A to E)
1
1
3
D
E
A B
C
1
2
• Optimal solution to the original problem (going from E to A)
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So what is all the fuss about DP?
• The examples we have seen so far are extremely simple (in fact naïve)
• There are major complications
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A (very) Famous Example
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Task: Move pieces from left to right
Rules• One piece at a time• Large on small is not allowed
• Objective ??
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Analysis
Analysis
Given n pieces• Move n-1 pieces to the intermediate position• Move the bottom one to the destination• Move the n-1 pieces from the intermediate
position to the destination
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Formalism
Formalism
Let S(n,x,y) := Solution to a problem involving
moving n pieces from position x to position y.
Thus, S(1,x,y) := Move a piece from x to y.Theorem:• S(n,x,y) = S(n-1,x,~(x,y)) , S(1,x,y) , S(n-
1,~(x,y),y)
• Length [S(n,x,y)] = 2n - 1
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Start
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2n - 1 = 24 1 = 16 1 = 15
15
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2n - 1 = 24 1 = 16 1 = 15
15
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15
2n - 1 = 24 1 = 16 1 = 15
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22nnThe Curse of
DimensionalityBellman [1957]
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What’s next?
• A short visit to the tutOR site.• Hands-on with the Knapsack
Problem:– Conventional Formulation– DP functional equation