Space Engineering pp 369-400 | Cite as

# Balance Layout Problems: Mathematical Modeling and Nonlinear Optimization

## Abstract

The paper studies the optimal layout problem of 3D-objects (solid spheres, straight circular cylinders, spherocylinders, straight regular prisms, cuboids and tori) in a container (a cylindrical, a parabolic, or a truncated conical shape) with circular racks. The problem takes into account a given minimal and maximal allowable distances between objects, as well as, behaviour constraints of the mechanical system (equilibrium, moments of inertia and stability constraints). We call the problem the Balance Layout Problem (BLP) and develop a continuous nonlinear programming model (NLP-model) of the problem, using the *phi*-function technique. We also consider several BLP subproblems; provide appropriate mathematical models and solution algorithms, using nonlinear programming and nonsmooth optimization methods, illustrated with computational experiments.

### Keywords

Layout problems Behaviour constraints Distance constraints*Phi*-functions Quasi-

*phi*-functions NLP-models Optimization algorithms

### References

- 1.Fasano, G., Pintér, J. (eds.): Modeling and Optimization in Space Engineering. Series: Springer Optimization and Its Applications. vol. 73, XII, 404 pp. Springer, New York (2013)Google Scholar
- 2.Fasano, G., Pintér, J. (eds.): Optimized Packings and Their Applications. Springer Optimization and Its Applications, vol. 105, 326 pp. Springer, Berlin (2015)Google Scholar
- 3.Che, C., Wang Y., Teng, H.: Test problems for quasi-satellite packing: Cylinders packing with behaviour constraints and all the optimal solutions known. Opt. (2008). Online http://www.optimization-online.org/DB_HTML/2008/09/2093.html
- 4.Lei, K.: Constrained layout optimization based on adaptive particle swarm optimizer. In: Zhihua, C., Zhenhua, L., Zhuo K., Yong, L. (eds.) Advances in Computation and Intelligence, vol. 1, pp. 434–442. Springer, Berlin (2009)CrossRefGoogle Scholar
- 5.Sun, Z., Teng, H.: Optimal layout design of a satellite module. Eng. Optim.
**35**(5), 513–530 (2003)CrossRefGoogle Scholar - 6.Jingfa, L., Gang, L.: Basin filling algorithm for the circular packing problem with equilibrium behavioural constraints. Science China Inf. Sci.
**53**(5), 885–895 (2010)CrossRefGoogle Scholar - 7.Oliveira, W.A., Moretti, A.C., Salles-Neto, L.L.: A heuristic for the nonidentical circle packing problem. Anais do CNMAC,
**3**, 626–632 (2010)Google Scholar - 8.Xu, Y.-C., Xiao R.-B., Amos, M.: A novel algorithm for the layout optimization problem. In: Proceedings of 2007 IEEE Congress on Evolutionary Computation (CEC07), pp. 3938–3942. IEEE Press, New York (2007)Google Scholar
- 9.Chazelle, B., Edelsbrunner, H., Guibas, L.J.: The complexity of cutting complexes. Discret. Comput. Geom.
**4**(2), 139–181 (1989)MathSciNetCrossRefMATHGoogle Scholar - 10.Kovalenko, A., Romanova, T., Stetsyuk, P.: Balance layout problem for 3D-objects: mathematical model and solution methods. Cybern. Syst. Anal.
**51**(4), 556–565 (2015). doi: 10.1007/s10559-015-9746-5 MathSciNetCrossRefMATHGoogle Scholar - 11.Chernov, N., Stoyan, Y., Romanova T.: Mathematical model and efficient algorithms for object packing problem. Comput. Geom. Theory Appl.
**43**(5), 533–553 (2010)MathSciNetCrossRefMATHGoogle Scholar - 12.Chernov, N., Stoyan, Y., Romanova, T., Pankratov, A.:
*phi*-functions for 2D objects formed by line segments and circular arcs. Adv. Oper. Res. (2012). doi: 10.1155/2012/346358 - 13.Stoyan, Y., Pankratov, A., Romanova, T.: Quasi-
*phi*-functions and optimal packing of ellipses. J. Global Optim. (2015). doi: 10.1007/s10898-015-0331-2 MATHGoogle Scholar - 14.Wachter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program.
**106**(1), 25–57 (2006)MathSciNetCrossRefMATHGoogle Scholar - 15.Shor, N.Z.: Nondifferentiable Optimization and Polynomial Problems, vol. 394. Kluwer Academic Publishers, Boston (1998)CrossRefMATHGoogle Scholar
- 16.Shor, N.Z., Stetsyuk, P.I.: Modified
*r*-algorithm to find the global minimum of polynomial functions. Cybern. Syst. Anal.**33**(4), 482–497 (1997)MathSciNetCrossRefMATHGoogle Scholar - 17.Stetsyuk, P., Romanova, T., Scheithauer, G.: On the global minimum in a balanced circular packing problem. Optim. Lett. (2015). doi: 10.1007/s11590-015-0937-9 MathSciNetMATHGoogle Scholar
- 18.Bennell, J., Scheithauer, G., Stoyan, Y., Romanova, T., Pankratov, A.: Optimal clustering of a pair of irregular objects. J. Glob. Optim.
**61**(3), 497–524 (2015)MathSciNetCrossRefMATHGoogle Scholar