# Introduction to Graph Theory

May 5th, 2015
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Operations research An introduction to solution methods Ecole des Mines de Nantes Master MOST 2012-2013

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Introduction to Graph TheoryOperations researchAn introduction to solution methodsEcole des Mines de Nantes Master MOST 2012-2013Olivier Pton- 1-ProblemMin f ( x )xS An optimization problem S is the solution set that represents all feasible solutions of a problem. f is the objective function that maps S to R. It evaluates each feasible solution. Also called evaluation function or cost function Minimization = maximization !max f ( x) min ( f ( x))xS xS- 2-Mathematical modeling1.Decision Variables x1,,xn A solution = a value for each variable Objective function min f(x1,,xn) Constraints g1(x1,,xn) a g2(x1,,xn) b g3(x1,,xn) = c2.3.- 3-Solutions of an optimization problem What is a feasible solution ? What is an optimal solution ? How many optimal solutions are there for a given problem ?- 4-Combinatorial Optimization (C.O.) problemsDifferent fields in Optimization: Linear Programming Non-Linear Optimization Integer Mixed Linear Programming Graph / network optimization Routing, Scheduling, Supply Chain,Combinatorial optimization studies optimization on finite and discrete domains.Find the minimum s* of f on a finite set S.f ( s ) Min f ( s)sS- 5-Characteristics of C.O. problems ?A solution is a combination of values given to different variables. The variables are discrete. The number of p

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