Mathematical modeling To describe or represent a realworld

Mathematical modeling • To describe or represent a real-world situation quantitatively, in mathematical language

Math models • Simplification- no model reproduces every property or aspect of the thing modeled. • Every model necessarily simplifies, to some extent, the reality it describes. • A useful model is simple enough to permit calculations but detailed enough to capture the essence of the situation being modeled.

prediction • A useful mathematical model should do more than describe already observed behavior. It should also help us predict or understand the behavior.

The process of modeling • Description- using mathematical ingredients, functions, variables, equations, etc. • Deduction- determining mathematical consequences of the model’s description • Interpretation- forming conclusions based on results of the model

Discrete Math • Methods of problem solving • Real-life mathematics • Topics such as scheduling, social decision making, modeling with graphs, counting techniques, optimization, cost effectiveness

Three main themes • Existence: Is there a solution? • Counting: How many solutions are there? • Efficiency: What is the best solution?

Hamilton circuits • A circuit containing every vertex of a graph • Visit each vertex (point) exactly once • Do not have to visit all edges (sides) • Must return to the starting vertex • A graph with a Hamilton circuit is called Hamiltonian

Does a graph have a Hamilton circuit? • There is no simple way to tell • The problem of determining if a graph is Hamiltonian is NP-complete. • This means that the number of steps needed to solve the problem, with the best known algorithms today, is exponential in the number of vertices in the graph.

• After a meeting at her home, Kaitlin has agreed to drive home Mary, Rachel, and Leslie. If the time (minutes) to drive between her friend’s homes is shown in the figure, what route gets Kaitlin back home the quickest? M 8 6 5 R 9 K 13 12 L

• The numbers on the graph are called weights • these can represent time, cost, or distance • a minimum-cost Hamiltonian circuit is one with the lowest possible sum to the weights of its edges

Algorithm • Generate all possible Hamiltonian tours • Add up the distances on the edges of each tour • Choose the tour of minimum distance

• The graph model is called a complete graph if the edge between any pair of vertices is present in the graph. • In the graph with 4 vertices, there would be six different ways to traverse a circuit, but half of them would be the reversal of the others. • In a complete graph with ‘n’ vertices, there are (n-1)!/2 ways to route a circuit.

TSP • This task of finding the shortest route is one of the most common in operations research, the branch of mathematics concerned with getting governments and businesses to operate more efficiently. It is usually called the traveling salesman problem (TSP), because of its early formation…. the trip of minimum cost that a salesperson can make to visit the cities in a sales territory and end up in the same city.

TSP problems • Optimization problems of this type are becoming increasingly more important in the world of communications, airline networking, wiring, networks, etc. . .

algorithms • Brute force- find all possible circuits and total each to find the optimum. • Nearest neighbor- start at a vertex, . . . at each step, go to the nearest vertex which has not yet been visited and will not close a circuit, when all have been visited, return to the starting point. • Cheapest link-pick the shortest edge, such that no circuit will be formed, and no vertex is greater than 2…. . repeat until closed.

• Find the minimum weight Hamilton circuit in the graph below. 3 1 2 5 6 4

Arc. View • GIS software- geographical information system • a great tool to create a model of TSP! • Topic--Using Arc. View, create a graph to model the transportation of students between high schools and look for an efficient route.