Introductory material Jonathan Godfrey

Introduction In this course we build upon ideas taught in , and you may benefit from past enrolment in other Decision Science, Statistics or Maths papers focused on linear programming ideas, and found optimal solutions to problems. Not all problems lend themselves to LP though and many do not have optimal solution techniques that are practical.

Problems with easily found optimal solutions Network problems Minimum cost. Maximum flow problems. Transportation or transshipment problems. In such work, we use algorithms.

Problems not solved easily Travelling Salesperson Problem The knapsack problem. Some integer programming problems. Some mixed problems with both continuous and integer valued variables.

A simple example Say we want to have the least number of coins floating around our pockets and we’re about to be given $3.65 in change. We’d expect to get $2, $1, 50c, 10c, and 5c coins handed to us. This is an example of a “Greedy” approach to problem solving. The largest possible coins were used initially to make the change.

The Greedy Approach Greedy algorithms attack a problem by constructing the complete solution in a series of steps. The general idea is to assign the values for all of the decision variables one at a time. At each step, the algorithm makes the best available decision, the ‘best’ profit, making it ‘greedy’. Unfortunately making the best decision at each step does not necessarily result in the best solution overall.

A Simple Example cont’d What would happen if we lived in a silly country that had no 5c coins, but did have 7c and 1c coins. The last 15c of our change would be given as 10c + 5×1c using the greedy approach. The optimal solution would actually be 2×7c and 1c instead.

A Simple Example cont’d The “Greedy” approach hasn’t worked. I’ve chosen the word “approach” here on purpose. Many authors would have used “Greedy algorithm”.

Algorithms and Heuristics We need to think carefully about what an approach actually achieves. An algorithm follows a step by step process until an optimal solution is found. Heuristics, on the other hand, just find a solution. Sometimes an algorithm is described as being heuristic. This course looks at many algorithms and heuristics to solve or find solutions to some (hopefully) interesting problems.