2. 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 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.

3. 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

4. Example jobPay ($)Time (min)Fun index 1133 2252 3122 4321 Maximize fun time so that you make at least $75 and do not spend more than two hours.

5. Example Formulation jobPayTimeFun 1133 2252 3122 4321 Only two variables are likely to be non-zero. Which do you expect them to be?

6. Continuous solution from Solver Without integer constraint, solution does not have to be integer Objective Cell (Max) CellNameOriginal ValueFinal Value $E$1fun80112.4999958 Variable Cells CellNameOriginal ValueFinal ValueInteger $B$1x1100Contin $B$2x2100Contin $B$3x31052.49999759Contin $B$4x4107.500000626Contin Constraints CellNameCell ValueFormulaStatusSlack $E$2pay74.99999946$E$2>=75Binding0 $E$3time119.9999964$E$3<=120Not Binding3.57628E-06

7. Integer solution Will rounding work? Objective Cell (Max) CellNameOriginal ValueFinal Value $E$1fun112.4999958112.0000005 Variable Cells CellNameOriginal ValueFinal ValueInteger $B$1x102.000000156Contin $B$2x200Contin $B$3x352.4999975949Integer $B$4x47.5000006268Integer Constraints CellNameCell ValueFormulaStatusSlack $E$2pay75.00000016$E$2>=75Binding0 $E$3time120.0000005$E$3<=120Binding0 $B$3=Integer $B$4=Integer

8. 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.