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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
