Discrete Structural Optimization by W. Gutkowski (eds.)

By W. Gutkowski (eds.)

The engineering layout of buildings and machines is composed frequently find the easiest answer between a finite variety of possible judgements. This quantity includes difficulties and answer equipment for discrete structural optimization. targeted, approximate and heuristic equipment are provided employing deterministic and stochastic approaches.

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Branch the initial problem in the two following subproblems f(x) (1) minimize subject to h(x) g(x) ~ 0; x~e ~ [x~0 )] g(x) ~ 0; Xk f(x) (2) minimize subject to = 0; h(x) = 0; ~ [xi0 >] + 1 and solve them as continuous problems. With the space [x~0 )] < Xk < [xi0>] + 1 all integer values of x~e are removed from the problem without removing integer values of remaining variables. Step 4. Continue branching the initial problem for all other non integer design variables x;. Some of obtained solutions may be integer and feasible.

Starting from A we can go first to C or D. It would be possible to make a right decision knowing the minimum distances from C to B and from D to B. Let denote the minimum cost from C to B by Sc and minimum from D to B by Sv. It is important to note that only travelling the best path from C and from D to B are essential to the above computations. The costs along the nine inferior paths from C and D to B need never to be computed. The mentioned above principle of optimality can be stated nowas follows : "The best path from A to B has the property, that whatever the initial decision at A, the remaining path to B, starting from next point after A, must be the best path from that point to B".

8. N. Distefano, A. Rath, A dynamic programming approach to the optimization of elastic trusses. J. optimization Theory and Application, vol. 15, 1975, no 1, pp. 13-26. 9. S. E. Dreyfus, Dynamic Programming and the Calculus of Variations, Academic Press, Inc. New York, 1965. 10. R. S. Garfinkel and G. L. Nemhauser, Integer Programming, Wiley, New York, 1972. Design Problems and Exact Solution Methods 11. E. Gomory, Outline of an algorithm for integer solutions to linear programs, Bull. Amer. Soc.

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