Integer linear programming
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Transcript of Integer linear programming
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Integer linear programmingOptimization problems where design variables have to be integers are more difficult than ones with continuous variables.The degree of difficulty is particularly damaging for large number of variables:With continuous variables finding a local optimum increases linearly or quadratically with the number of variables.With integer variables it can increase exponentially (or non-polynomially): These problems are NP-hard.Problems with linear objective and linear constraints (Linear programming) are easier to solve.Furthermore: They do not have optima that are local but not global.
FormulationThe standard form of an integer programming problem is
Algorithm papers often limit themselves to standard form, but software usually allows the more general form
ExamplejobPay ($)Time (min)Fun index1133225231224321Maximize fun time so that you make at least $75 and do not spend more than two hours.
Example Formulation
jobPay Time Fun 1133225231224321Only two variables are likely to be non-zero. Which do you expect them to be?
Continuous solution from SolverWithout integer constraint, solution does not have to be integerObjective Cell (Max)CellNameOriginal ValueFinal Value$E$1fun80112.4999958Variable CellsCellNameOriginal ValueFinal ValueInteger$B$1x1100Contin$B$2x2100Contin$B$3x31052.49999759Contin$B$4x4107.500000626ContinConstraintsCellNameCell ValueFormulaStatusSlack$E$2pay74.99999946$E$2>=75Binding0$E$3time119.9999964$E$3=75Binding0$E$3time120.0000005$E$3