lec10 - IE521 Advanced Optimization Lecture 10 Dr Zeliha...

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IE521 Advanced Optimization Lecture 10 Dr. Zeliha Akc ¸a December 2011 1 / 27
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Integer Programming I An integer programming model is a mathematical programming model in which some decision variables are integers. I Note that we are interested in integer programming models in which objective function and constraint functions are linear! min c > x s.t. Ax b x Z n where A R mxn , c R n , b m . I Why do we need to consider decisions that should be integer? I When the decision variable is associated with an entity that is indivisible, we need integer variables. I For example, number of aircraft to purchase, number of facilities to open, or vehicles to schedule. 2 / 27
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Integer Programming Classes 1. Pure integer programming problem: min c > x s.t. Ax b x Z n 2. Midex integer programming problem: min c > x + h > y s.t. Ax + Gy b x Z n y R p 3. 0-1 integer programming problem: min c > x s.t. Ax b x ∈ { 0 , 1 } 3 / 27
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Combinatorial Program I Combinatorial problem is a special case of integer programming problem. I Let N = { 1 , 2 , .., n } . I Consider a finite subsets of set N . I These subsets are the solutions to the integer program and there is an objective function value for each. I vehicle routing problem, traveling salesman problem, assignment problem, knapsack problem are some examples of combinatorial problems. 4 / 27
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Programming With Integers I Why do we use 0-1 (binary) variables? I We use binary variables for many purposes: To model yes - no decisions To enforce disjunctions (either ( x S ) or ( x K ) To enforce logical conditions To model with fixed costs To model piecewise linear functions I We need integer programming when the linear approximation of the decision is not true in practice 5 / 27
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LP relaxation I Linear relaxation of an integer program: IP Model: min z = c > x s.t. Ax b x Z n + LP relaxation: min z LP = c > x s.t. Ax b x 0 I If x ∈ { 0 , 1 } then the LP relaxation will include 0 x 1 I An LP relaxation of an integer program is a lower bound for the model: z * LP z * where z * LP is the optimal solution to LP relaxation, z * is the optimal solution to IP I The LP relaxation is a lower bound for the optimal IP solution (if maximization problem, then the LP is an upper bound). 6 / 27
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LP Relaxation, Cont. I When we solve the LP relaxation, and the variables that needs to be integer are integer, then it is also the optimal solution to the IP. I Can we solve LP relaxation and round the fractional variables in order to find the optimal solution? I
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This note was uploaded on 01/09/2012 for the course IE 521 taught by Professor Zelihaakça during the Fall '11 term at Fatih Üniversitesi.

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lec10 - IE521 Advanced Optimization Lecture 10 Dr Zeliha...

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