Integer linear programming Optimization problems where design variables

Integer linear programming • Optimization 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 nonpolynomially): 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.

Formulation • The standard form of an integer programming problem is • Algorithm papers often limit themselves to standard form, but software usually allows the more general form

Example job Pay ($) Time (min) Fun index 1 1 3 3 2 2 5 2 3 4 1 3 2 2 2 1 Maximize fun time so that you make at least $75 and do not spend more than two hours.

Example Formulation Only two variables are likely to be non-zero. Which do you expect them to be? job Pay Time Fun 1 1 3 3 2 2 5 2 3 1 2 2 4 3 2 1

Continuous solution from Solver • Without integer constraint, solution does not have to be integer Objective Cell (Max) Cell Name Original Value Final Value $E$1 fun 80 112. 4999958 Variable Cells Cell Name Original Value Final Value Integer $B$1 x 1 10 0 Contin $B$2 x 2 10 0 Contin $B$3 x 3 10 52. 49999759 Contin $B$4 x 4 10 7. 500000626 Contin Constraints Cell Name $E$2 pay $E$3 time Cell Value Formula 74. 99999946 $E$2>=75 119. 9999964 $E$3<=120 Status Binding Not Binding Slack 0 3. 57628 E-06

Integer solution • Will rounding work? Objective Cell (Max) Cell Name Original Value Final Value $E$1 fun 112. 4999958 112. 0000005 Variable Cells Cell $B$1 $B$2 $B$3 $B$4 Name Original Value Final Value Integer x 1 0 2. 000000156 Contin x 2 0 0 Contin x 3 52. 49999759 49 Integer x 4 7. 500000626 8 Integer Constraints Cell Name $E$2 pay $E$3 time $B$3=Integer $B$4=Integer Cell Value Formula 75. 00000016 $E$2>=75 120. 0000005 $E$3<=120 Status Slack Binding 0

Integer problems in laminate design • When ply angles are unrestricted we have he restriction of integer number of plies. • Most points on the Miki diagram are accessible, but with specific ply angles. • When angles are limited to a small set, together with integer number of plies, we are limited to a finite number of points on diagram. • We will investigate which laminate design problems can be cast as linear integer programming. • Some times it requires some ingenuity.