1. Integer Programming and Logic-Based Modeling Jan Fábry, Jan Pelikán ___________________________________________________________________________ MME 2003, Prague

2. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Integer Programming Models Discrete variables: NP-hard problems  Non-polynomial method (exponential number of branches)  Estimation of the optimal objective value Standard Branch and Bound Method

3. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Integer Programming Models Reduction of number of branches  Skiping non-effective branches (using the estimation of the optimal objective value)  Heuristic methods  Logic-based Branch and Bound Method

4. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague TSP – IP Model

5. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague TSP – Logic-Based Model

6. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Integer Knapsack Problem Example: 3 objects Cost = (5, 8, 4) Profit = (3, 5, 2) Value  {1, 2, 3, 4} All different Feasible (4, 2, 1) Infeasible (2, 2, 1) Objective: minimize total cost Total profit  30

7. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Integer Knapsack Problem IP modelLogic-based model

8. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Logic-based Approach  Reduction of branches Advantage  Simplicity of L-B models Disadvantage  No estimation of the optimal value

9. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Integer Knapsack Problem Domain VARVALUE x1x1 1234 x2x2 1234 x3x3 1234 VARVALUE x1x1 2 x2x2 4 x3x3 3 VARVALUE x1x1 2 x2x2 2 x3x3 3

10. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Integer Knapsack Problem Domain reduction  x1x1 12 x2x2 12 x3x3 34

11. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Integer Knapsack Problem Bounds consistency maintenance  x1x1 1234 x2x2 234 x3x3 1234

12. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Integer Knapsack Problem 1234 234 1234 2 4 3 34 234 1234 x 1 ≤ 2x 1 ≥ 3 z = 54 4 3 2 3 4 12 x 2 ≤ 3x 2 ≥ 4 z = 52 3 4 2 3 4 1 x 3 ≤ 1x 3 ≥ 2 z = 51z = 55

13. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Logic-Based Modeling Defined predicates all-different {y 1,…,y n }  The domains of variables y j may have more than n elements.  The constraint requires that variables y 1,…,y n take distinct values.  It is mostly used in assignment problems.

14. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Logic-Based Modeling Defined predicates circuit (y 1,…,y n )  The constraint requires that y j is the integer occurring after j in some permutation of 1,2,…,n.  Used in vehicle routing applications. 

15. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Logic-Based Modeling Defined predicates path (y 1,…,y k ) 

16. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Logic-Based Modeling Defined predicates distribute ((m 1,…,m n ), (x 1,…,x n ), a)  The constraint requires that value of x j occurs exactly m j times in array a of an arbitrary length. 

17. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Logic-Based Modeling Defined predicates cumulative ((t 1,…,t n ), (d 1,…,d n ), (c 1,…,c n ), C)  Used in job scheduling problems.  Variable t j represents the start time of the job j.  Parameters d j and c j are the duration and the consuming resource rate of the job j.  The resource limit is C.

18. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Logic-Based Modeling Defined predicates element (y, (c 1,…,c n ), z)  The constraint requires that z = c y.  y and z are variables.  All c j are constants.  Used in lot-sizing problems.

19. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Software Standard Branch and Bound Method LINGO, AMPL, XPRES-MP, ILOG CPLEX,… Logic-based modeling CHIP V5, OPBDP, ILOG Solver (OPL Studio),… TRIAL VERSION: http://www.ilog.com/products/oplstudio/trial.cfm

20. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Mathematical Recreation JAN DNES NEVI ZDA SE JEDE JEDNA JIZDA ZNOVA enum Letters {J,A,E,N,D,S,V,I,Z,O}; var int k[Letters] in 0..9; solve { alldifferent(k) onDomain; 100*k[J]+10*k[A]+k[N] +1000*k[D]+100*k[N]+10*k[E]+k[S] +1000*k[N]+100*k[E]+10*k[V]+k[I] +100*k[Z]+10*k[D]+k[A] +10*k[S]+k[E] +1000*k[J]+100*k[E]+10*k[D]+k[E] +10000*k[J]+1000*k[E]+100*k[D]+10*k[N]+k[A] +10000*k[J]+1000*k[I]+100*k[Z]+10*k[D]+k[A] =10000*k[Z]+1000*k[N]+100*k[O]+10*k[V]+k[A]; k[J]>=1; k[D]>=1; k[N]>=1; k[Z]>=1; k[S]>=1; [J]<=5; };

21. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Mathematical Recreation JAN DNES NEVI ZDA SE JEDE JEDNA JIZDA ZNOVA 201 3189 1854 630 98 2838 28310 24630 61750  10 solutions

22. ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague Questions