ALGORITHM SYSTEMS FOR COMBINATORIAL OPTIMIZATION: HIERARCHICAL MULTISTAGE FRAMEWORK

Transcription

1 ALGORITHM SYSTEMS FOR COMBINATORIAL OPTIMIZATION: HIERARCHICAL MULTISTAGE FRAMEWORK Dr. Mark Sh. Levin, The Research Inst., The College Judea & Samaria, Ariel, Israel Introduction In recent decades, signicance heuristic approaches to combinatorial optimization s is increasing. The situation is based on the fact that majority real applied s are very complicated and require composite heuristic solving schemes which can be considered as hierarchical multi-stage algorithm systems. Thus it is reasonable to examine systems engineering approach to a design the algorithm systems. The list traditional well-known heuristic approaches involves the following [1, 2, 4, 5,, 7, 8, 9, 10, 11, 14, 15, 17, 18, 19, 21, etc.]: (1) approximation solving schemes; (2) genetic algorithms and evolutionary computing; (3) tabu-search; (4) simulated annealing; (5) greedy algorithms; () constraint satisfaction approach; and (7) probabilistic approaches. In our opinion, a basic trend in the eld heuristic design for combinatorial optimization is the following: FROM algorithms [1, etc.] TO algorithm frameworks [1, etc.]. A three-stage algorithm framework is described for a sport timetabling by Nemhauser and Trick [1]. Multi-stage algorithm systems have been proposed in works Glover et al. [5,, etc.]. Moreover, the heuristic algorithm systems can be considered as specic solving environments which have been studied in recent years [3, etc.]. In the paper, a structure the systems, kinds and basic algorithm frameworks are proposed which can be considered as a step to the design the algorithm systems. Hierarchy System The essense the contemporary novelty in this eld is based on the following: eectiveness simple algorithmic schemes is limited and it is reasonable to build more complex multi-level, hierarchical solving frameworks which involve series-parallel and adaptive solving. A structure the algorithm system can be analyzed as follows: 1. Control unit (planning, adaptation): (a) selection corresponding basic algorithms, and (b) design composite solving strategy. 2. Level execution: (a) analysis an initial, (b) execution a solving, and (c) analysis. 3. Level bases / repositories: (a) s and examples, (b) algorithms and procedures, and (c) solving strategies.

2 Kinds System Structure Here let us examine basic kinds algorithm systems structures. First, let us point out main parameters the algotihm systems: 1. Kind algorithm base: (a) one-algorithm framework; (b) algorithm set with parallel execution the algorithms; and (c) complex multistage algorithm framework (e.g., series, series-parallel, etc.). 2. Feedback: (a) without feedback; (b) with feedback two types as follows: (i) analysis resultant solution; and / or (ii) analysis intermediate. 3. Kind algorithm framework (scheme, composite strategy) synthesis: (a) xed algorithm framework (algorithm strategy); (b) exible design algorithm framework (e.g., selection standard algorithm strategies, synthesis new algorithm frameworks). Thus the following functional operations have to be examined: (1) analysis (input), (2) solving, (3) analysis, (4) modication / selection algorithm, (5) selection / synthesis an algorithm set, () aggregation, and (7) selection / synthesis an algorithm scheme. Clearly, the simplest solving scheme involves initial, solving, result, and result analysis). Figs. 1, 2, 3, and 4 depict possible more complicated situations (e.g., parametric modication / selection algorithm, selection / design algorithm scheme). Initial - Modication algorithm (algorithm) Fig. 1. Simple one-algorithm framework (output) - Initial - Modication & selection algorithm (algorithm) (output) - Repository s Fig. 2. One-algorithm framework with algorithm base Note it is possible to add into the multi-algorihtm framework at Fig. 2 a block for aggregation which were obtained on the basis dierent algorithms.

3 Initial - Modication & selection algorithms (algorithms) (output) - Repository s Fig. 3. Multi-algorithm framework with algorithm base Initial - Design algorithm scheme (scheme) (output) - Repository s Scheme Methods for scheme synthesis Fig. 4. Multi-algorithm scheme framework Some Frameworks Our list the some prospecitve solving schemes (frameworks) consists the following non-typical schemes and/or models: Framework 1. Hierarchical morphological approach on the basis morphological clique including steps as follows [12]: (1.1) decomposition, (1.2) searching for multialternative local decisions, and (1.3) integration local decisions into global decisions for the initial. Framework 2. Multi-level framework on the basis k-exchange techniques [19, 20]: (2.1) hierarchical base k-exchange heuristics; (2.2) special control unit for the following: (i) analysis an individual, (ii) planning the solving (selection algorithms and synthesis solving strategy); and (2.3) analysis intermediate and correction solving. Framework 3. Repository approach [13]: (3.1) model repository; (3.2) analysis and approximation initial on the basis repository; and (3.3) selection and/or design algorithmic schemes. Framework 4. Spacelling curves approach including their generation on the basis various

4 kinds fractals [2, etc.]. Here we do not examine issues special decomposition approaches. We guess the use some well-known decomposition algorithms (e.g., clustering) will be sucient. The stage planning (design solving composite strategies) has to be based on multicriteria decision making and system integration methods which are basic ones in systems engineering [12, etc.]. Table 1 depicts correspondence the frameworks and classes applied s. Framework Morphological approach Framework on the basis k-exchange technique Framework on the basis repositories Spacelling curves Table. 1. Frameworks and Problems Classes Combinatorial Problems Scheduling TSP Minimal Problem Steiner Tree Problem * * * * * * * * Allocation/ Assignment Problem * Example Series-Parallel Framework Here an example series-parallel solving strategy is described (multicriteria ranking alternatives) [12]. The approach was implemented in DSS COMBI [12, etc.]. In the main, an information part decision support system involves the following components: (1) (alternatives, criteria, multicriteria estimates alternatives upon criteria, preference relations); and (2) tools for management. We consider the following types information: (mainly, preference relations) R j (j = 0; :::3): (1) initial information as alternatives, criteria, multicriteria estimates (R 0 ); (2) preference relation on alternatives (R 1 ); (3) intermediate linear ordering alternatives (R 2 ); (4) ranking alternatives (R 3 ) i.e., generation a priority for each alternatives. The following ing operations are the basic ones: (i) basic ing; (ii) aggregation ; and (iii) parallelization solving. Thus we point out the following hierarchy the solving : (1) algorithms and /or interactive procedures; (2) series-parallel composite strategies (including series-parallel) (3) scenaria (strategy framework that includes stages result analysis and feedback too). The composite strategy for multicriteria ranking consists the following three stages: R 0 =) R 1 =) R 2 =) R 3 or S = H T U where H corresponds to the design R 1 : T corresponds to the design R 2 ; and U corresponds to the design R 3. Considered basic local techniques (algorithms and procedures) are the following:

5 Stage H: pairwise comparison (H 1 ); ELECTRE-like technique (H 2 ); technique additive utility function (H 3 ); and expert stratication (H 4 ). Stage T : line elements sum preference matrix (T 1 ); technique additive utility function (T 2 ); series revelation maximal elements (T 3 ); series revelation Pareto-eective elements (T 4 ); and expert stratication (T 5 ). Stage U: series revelation maximal elements (U 1 ); series revelation Pareto-eective elements (U 2 ); dividing linear ranking (U 3 ); and expert stratication (U 4 ). The design morphological space and an example series-parallel composite strategy are depicted in Fig. 5. Solving strategy S = H T U S 1 &S 2 &S 3 = (H 4 T 5 U 4 )&(H 2 T 1 U 3 )&(H 2 T 3 U 1 ) Generation preference relations R 1 x Generation linear ordering R 2 Ranking R 3 e e e H 1 T 1 U 1 H 2 H 3 H 4 H T U T 2 T 3 T 4 T 5 Fig. 5. Resultant design morphology U 2 U 3 U 4 Conclusion We have examined solving frameworks for some combinatorial optimization s. It is reasonable to point out the following directions future research: 1. Design and analysis several non-standard heuristic solving schemes above including the selection and design composite solving strategies. 2. Application the above-mentioned heuristic solving schemes for well-known combinatorial optimization s (scheduling, TSP, assignment / allocation, etc.). 3. using the solving schemes above for multicriteria situations. 4. Studies some signicant applications (e.g., facilities design / planning, scheduling, network design, channel / frequency assignment, team design). The research was supported by the Center Absorption in Science, The Ministry Absorption, State Israel. References 1. Ball, M. and Magazine, M. (1981), "The Design and Heuristics", Networks, Vol. 11, No. 2, pp Bartholdi, J.J., III and Platzman, L.K. (1989), "Spacelling Curves and the Planar Travelling Salesman Problem", J. the ACM, Vol. 34, No. 4, pp Gallopoulos, E., Houstis, N.E. and Rice, J.R. (1995), "Workshop on Problem-Solving Enviroments: Findings and Recommendations", ACM Computing Surveys, Vol. 27, No. 2, pp

6 4. Garey, M.R. and Johnson, D.S. (1979), Computers and Intractability. The Guide to the Theory NP-Completeness, San-Francisco, W.H. Freeman and Company. 5. Glover, F., Klingman, D., Phillips, N. and Ross, T. (1985), "Integrating Modeling, Design and Computational Implementation to Solve Large Scale Non-Linear Mixed Integer Programming Problems", In Annals Operations Research, C. Monma, ed., Vol. 5, pp Glover, F. and Laguna, M. (1997), Tabu Search, Boston, Kluwer. 7. Goldberg, D.E. (1989), Genetic in Search, Optimization and Machine Learning, Mass: Addison-Wesley, Reading. 8. Himmelblau, D.M., ed. (1973), Decomposition Large-Scale Problems, New York, North-Holland. 9. Holland, J.H. (1975), Adaptation in Natural and articial Systems, Ann Arbor, Michigan, Univ. Michigan Press. 10. Karp, R. (1997), "Probabilistic Partitioning for the Traveling- Salesman Problem", Mathematics Operations Research, Vol. 2, No. 3, pp Laporte, G. and Osman, I., eds. (1995), Metaheuristics in Combinatorial Optimization, Annals Operations Research, Vol. 9, Amsterdam, Baltzer. 12. Levin, M.Sh. (1998), Combinatorial Engineering Decomposable Systems, Dordrecht, Kluwer. 13. Levin, M.Sh., Sethi, S. and Sriskandarajah, S. (1998), "Towards Assignment-like Facilities Layout Problem", Unpublished Manuscript, Univ. Toronto. 14. Mackworth, A.K. (1987), "Constraint Satisfaction", in Encyclopedia on Articial Intelligence, S.C. Shapiro, ed., New York, J.Wiley & Sons, pp Michalewicz, M.D. (199), Genetic s + Data Structure = Evolutionary Programs, 3rd ed., Berlin, Springer-Verlag. 1. Nemhauser, G.L. and Trick, M.A. (1998), "Scheduling a Major College Basketball Conference", Operations Research, Vol. 4, No. 1, pp Pearl, J. (198), Heuristics, Prentice-Hall. 18. Reeves, C.R. (1995), Modern Heuristic Techniques for Combinatorial Problems, New York, J.Wiley & Sons. 19. Reinelt, G. (1994), The Traveling Salesman, LNCS, Vol. 840, Berlin, Springer-Verlag. 20. Sidney, J.B. and Levin, M.Sh. (1998), "K-exchange Technique for Optimization Over Permutations", Working Paper No , Univ. Ottawa, Faculty Business Administration. 21. Ulungu, E.L. and Teghem, J. (1994), "Multi-objective Combinatorial Optimization Problems: A Survey", J. Multi-Criteria Decision, Vol. 3, No. 2, pp