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Integer Programming and Logic-Based Modeling Jan Fábry, Jan Pelikán

Integer Programming and Logic-Based Modeling Jan Fábry, Jan Pelikán. ___________________________________________________________________________ MME 2003, Prague. Integer Programming Models. Discrete variables: NP-hard problems.

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Integer Programming and Logic-Based Modeling Jan Fábry, Jan Pelikán

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

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

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

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

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

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

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

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

  9. VAR VAR VAR VALUE VALUE VALUE x1 x1 x1 1 2 2 2 3 4 x2 x2 x2 1 2 2 3 4 4 x3 x3 x3 1 2 3 3 3 4 Integer Knapsack Problem Domain ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague

  10. x1 1 2 x2 1 2 x3 3 4 Integer Knapsack Problem Domain reduction ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague 

  11. x1 1 2 3 4 x2 2 3 4 x3 1 2 3 4 Integer Knapsack Problem Bounds consistency maintenance ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague 

  12. 1 2 2 3 3 3 3 3 4 4 4 2 2 3 3 3 4 4 4 4 4 4 1 1 1 1 2 2 2 2 2 3 3 3 4 4 x1 ≤ 2 x1 ≥ 3 z = 54 x2 ≤ 3 x2 ≥ 4 z = 52 x3 ≤ 1 x3 ≥ 2 z = 51 z = 55 Integer Knapsack Problem ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague

  13. Logic-Based Modeling Defined predicates ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague all-different {y1,…,yn} • The domains of variables yj may have more than n elements. • The constraint requires that variablesy1,…,yntake distinct values. • It is mostly used in assignment problems.

  14. The constraint requires that yj is the integer occurring after j in some permutation of 1,2,…,n. • Used in vehicle routing applications. Logic-Based Modeling Defined predicates ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague circuit (y1,…,yn)

  15. Logic-Based Modeling Defined predicates ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague path (y1,…,yk)

  16. The constraint requires that value of xjoccurs exactly mj times in array a of an arbitrary length. Logic-Based Modeling Defined predicates ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague distribute ((m1,…,mn), (x1,…,xn), a)

  17. Logic-Based Modeling Defined predicates ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague cumulative ((t1,…,tn), (d1,…,dn), (c1,…,cn), C) • Used in job scheduling problems. • Variable tj represents the start time of the job j. • Parameters dj and cj are the duration and the consuming resource rate of the job j. • The resource limit is C.

  18. Logic-Based Modeling Defined predicates ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague element (y, (c1,…,cn), z) • The constraint requires that z = cy. • y and z are variables. • All cj are constants. • Used in lot-sizing problems.

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

  20. 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; }; ___________________________________________________________________________ Integer Programming and Logic-Based Modeling MME 2003, Prague

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

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

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