Building “ Problem Solving Engines ” for Combinatorial Optimization

1. Building“Problem Solving Engines”for Combinatorial Optimization Toshi Ibaraki Kwansei Gakuin University (+ M. Yagiura, K. Nonobe and students, Kyoto University) Franco-Japanese Workshop on CP, Oct. 25-27, 2004

2. Problem solving engines for discrete optimization problems Approaches to general solvers • Attempts from artificial intelligence GPS (general problem solver), resolution principle, ..., CP (constraint programming) • Attempts from mathematical programming Linear, nonlinear, integer programming, ...

3. 1. All problems in NP can be reduced to IP Complexity Theory • Class NP Contains almost all problems solvable by enumeration • NP-hard (NP-complete)SAT (satisfiability), IP (integer program), . . . • Two implications 2.No algorithm can solve IP in polynomial time

4. Approximate solutions are sufficient in most applications. NP-hard problems can be approximately solved in polynomial time. But . . . Approach by Approximate Solutions • Problem sizes may explode duringreduction processes. • e.g. the number of variables may become n2 or n3. • 2. The distance to optimality may not be preserved. • Good approximate solutions to IP may not be good solutions to the original problem. Only “natural” reductions are meaningful.

5. Approach by Standard Problems

6. List of Standard Problems • Integer programming (IP) • Constraint satisfaction problem(CSP) • Resource constrained project scheduling problem (RCPSP) • Vehicle routing problem (VRP) • 2-dimensional packing problem (2PP) • Generalized assignment problem (GAP) • Set covering problem (SCP) • Maximum satisfiability problem (MAXSAT)

7. Efficiency, generality, robustness, flexibility, . . . Can such algorithms exist? Local search (LS) Approximation Algorithms Yes! • Metaheuristics Genetic algorithm, simulated annealing, tabu search, iterated local search, GRASP, variable neighborhood search, . . .

8. Standard problem: Constraint satisfaction problem (CSP)

9. CSP: Definition • nvariablesXiand their domains Di • m constraints Clequalities, inequalities, nonequalities (all-different), linear and nonlinear formulae • Hard and soft constraints;weights wl given to constraints Cl • Minimization of total penaltｙ p(X) =Σwlpl(X) pl(X):penalties given to violations of Cl

10. Comparison with IP • Flexible forms of constraints Compact formulations with small numbers of variables and constraints • Soft constraints and objective functions via penalty functions • Algorithms by metaheuristics Robust performance even for problems not suited for IP

11. CSP Algorithm • Algorithm framework: tabu search • Local search using shift neighborhood Checks all solutions obtainable by changing the value of one variable • Tabu list Prohibits changing those variables whose values were modified in recent t iterations, where t is tabu tenure.

12. Improvements • Reduction of the neighborhood size Data structures to skipXi and their values having apparently no improvement (i.e. partial propagation) • Evaluation function for the search q(X) =Σvlpl(X) (possiblyvl≠wl) • Automatic control of weights vl Frequent violation of Cl largervlSimilar to subgradient method for Lagrangean multipliers

13. References for details • K. Nonobe and T. Ibaraki, A tabu search approach to the constraint satisfaction problem as a general problem solver, European J. of OR, Vol. 106, pp. 599-623, 1998. • K. Nonobe and T. Ibaraki, An improved tabu search method for the weighted constraint satisfaction problem, INFOR, Vol. 39, No. 2, pp. 131-151, 2001. • M. Fukumori, Tabu search algorithm for the quadratic constraint satisfaction problem, Master thesis, Kyoto University, 2004.

14. Formulation to CSP： Variables Xij（nurse i, j-thday） Domain Dij={D, E, N, M, OFF} CSP: Case studyNurse scheduling problem • 25 nurses （Team A:13, B:12） • Experienced nurses and new nurses • 3 shifts （day, evening, night）, meetings, days off • Time span： 30 days

15. Nurse scheduling problemConstraints • Required numbers of shifts D, E, N in each day • Upper and lower bounds on the numbers of shifts and OFF’s assigned to each nurse in a month • Predetermined M’s and OFF’s • At least one OFF and one D in 7 days • Prohibited patterns: 3 consecutive N; 4 consecutive E; 5 consecutive D; D, E or M after N; D or M after E; OFF-work-OFF • N should be done in the form NN; at least 6 days before the next NN • Balance between teams A and B • Many others

17. Formulation to CSP： Variables Xtj（t-thweek, group j） Domain D = power set of {i=1, 2, …, n} Nonlinear constraints CSP: Case studySocial golfer problem • n golfers play once a week, always in m groups, each consisting of n/m players. • No two golfers want to play together more than once. • Find a schedule with the largest number of weeks.

19. Future Directions • Further improvement of metaheuristic algorithms • Increasing the formulation power of standard problems • Other standard problems • Aggregation of all algorithms into a decision support system • User interfaces. Supports to model application problems