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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 Applied Maths OR Optimization Sequential 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) Tip Usually 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 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 Which one do you like better?

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Help Desk 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

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Beware of the distinction between Problem Formulation Describing what the problem is all about Problem Solving Constructing 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 This is the (maths) model

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Observations There are infinitely many feasible solutions: Example 2(w + h) = p = 6 w = 3 - h/2 We can let h take value in the range (0,3).

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Observations Not difficult to show that the optimal solution is I am a square Prove this!

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Strong Operations Research Flavour Warning Parental Guidance is required

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Role of Models in Problem Solving Realisation Informal Description Formal Model SolutionWorking System Modelling Analysis Implementation Monitoring mathematical ^ Don’t panic!

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In Real-life Realisation Informal Description Formal Model SolutionWorking System Modelling Analysis Implementation Monitoring

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We shall focus on Realisation Informal Description Formal Model Solution Working System Modelling Analysis Implementation Monitoring Given

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Beware of the distinction between Problem Formulation Describing what the problem is all about Problem Solving Constructing 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. 12345 678910 1112131415 1212 1 2 10 15 20 10 1212 1 15 20 10 1 2 1

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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, 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) = 12345 678910 1112131415 1212 1 2 10 15 20 10 1212 1 15 20 10 1 2 1

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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 1 3 D E AB 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 AB C 1 2 Optimal solutions to the FE (going from A to E) 1 1 3 D E AB 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 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 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)] = 2 n - 1

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2 n - 1 = 2 4 1 = 16 1 = 15 15

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2 n - 1 = 2 4 1 = 16 1 = 15 15

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2 n - 1 = 2 4 1 = 16 1 = 15

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2 n The Curse of Dimensionality Bellman [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

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