Week 9 Optimisation What is optimisation? Examples/uses from business & industry? What is the Travelling Salesman Problem? What is the best PROVEN solution?

(TSP) Travelling Salesman Problem: A travelling salesman has to visit a number of cities (all & once) then return home, like a tour. We want to find the shortest route. For two cities it’s trivial, go from A to B and back to A. There’s only one possible tour A B  A. For 3 cities it’s fairly easy A  B, B  C, C  A (but you can go ABCA or ACBA, they’re the same length, so they’re the same ‘tour’ i.e. there’s only 1 tour). How many tours are there for 4 cities? How many for 5? What’s the formula/equation? What is the highest PROVEN solution you can find for TSP?

TSP Calculate the number of possible solutions for 6 cities The Traveling Salesman Problem is typical of a large class of “hard” optimization problems. The problem quickly ‘explodes’ in difficulty, like a polynomial  It has applications in science, business and engineering. For example, in the manufacture of a circuit board, it is important to determine the best order in which a laser will drill thousands of holes. An efficient solution to this problem reduces production costs for the manufacturer. Calculate the number of possible solutions for 6 cities

The Knapsack problem

0-1 Knapsack The knapsack problem is a problem in combinatorial optimization. Given a set of items, each with a Cost and a Value, determine the number of each item to include in a collection so that the total cost is less than a given limit and the total value is as large as possible. A similar problem often appears in business, combinatorics, complexity theory, cryptography and applied mathematics. The decision problem form of the knapsack problem is the question “can a value of at least V be achieved without exceeding the cost C?” There are many examples from business & industry e.g. paper mills which produce huge rolls of paper and cut to order, customers want different size and length rolls… 1% wastage could equal $1M. http://en.wikipedia.org/wiki/Cutting_stock_problem Example in Australia, water management Scheduling What does the 0-1 mean?

Which items should be chosen to maximize the value while still keeping the overall weight under or equal to 15 kg?

The Knapsack applet (on the LMS)

Brute force or naïve method Pros and cons of this method?

The Genetic Algorithm What is a genotype? Crossover? Mutation?

The results

Genetics Ever heard of a Liger = tiger/lion (or a tigon). What does a 400kg liger eat for breakfast? …whatever he wants. Is this an example of crossover or mutation? How about this ‘peppered’ moth. It used to look like the one on the left, then the Industrial Revolution came along…

Crossover Mutation Or, 11001011 + 11011111 = 11001111 Take part of one ‘parent’ and part of the other and join them together. Or, 11001011 + 11011111 = 11001111 Mutation 11001111  11001110 

To do Knapsack exercise From last session… Data visualisation Namevoyager: http://domino.research.ibm.com/comm/wwwr_thinkresearch.nsf/pages/20050901_baby.shtml Is this data mining? Gapminder & Google ‘Motion Charts’ www.gapminder.org