# Combinatorial Optimization.pdf - Combinatorial optimization...

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So far, we mostly looked atnumerical optimization with floating point decision variables.These are more generic irrespective of the fields the problem is arising from.There are alsodiscrete optimization problemswhere the decision variables take integervalues. There are two types: Integer programming and combinatorial problems.Integer Programming:Integer decision variablesâ†’Numerical optimization withround/floor/ceiling.Combinatorial Problems:â€¢Traveling Salesmanâ€¢Schedulingâ€¢Timetablingâ€¢Routingâ€¢Packing/Cuttingand manymore â€¦â€¢Several problem specific objectives, constraints, etc.â†’requiring problem specific solutionproceduresâ€¦Combinatorial optimization Problems1
Traveling Salesman Problem2â€¢The most famous combinatorial optimization problem is theâ€¢Traveling Salesman Problem(TSP).â€¢Given a list ofncities and distances between each pair of cities,â€¢the objective of the TSP is to construct a closed tourâ€¢Visiting each city just once with minimum total distance.â€¢The TSP is an example of anNP-completehardcombinatorial problemâ€¢Online resources available for TSP, e.g., software, etc.â€¢There are numerous TSP variants â€¦â€¢NP: Non-deterministic Polynomial-time.Complexity of solving theseproblems increases much faster than polynomial (i.e., exponential) withincreasing size of the problem.
Cutting Stock Problem3â€¢TheCutting Stock Problemis an NP-complete optimization problem that arises inmany applications in industry.â€¢The classicone-dimensionalcutting stock problem is to determine how to cut rolls ofpaper of fixed-width into customer orders for smaller widths so as to minimize waste.â€¢Many variants of this problem existâ€¦Packing Problemâ€¢Packing Problem(complementary to cutting problems) is to fill a larger space withspecified smaller shapes in the most economical (profitable) way.â€¢There aregeometric packing problemsin one dimension (1D), 2D and 3D,â€¢3D packing are those arise in filling delivery trucks or shipping containers.

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