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Russia Yekaterinburg
Section  Mathematics 
Title  Local dynamic programming incuts in routing problems with restrictions 
Author(s)  Petunin A.A.^{a}, Chentsov A.G.^{ab}, Chentsov P.A.^{b} 
Affiliations  Ural Federal University^{a}, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences^{b} 
Abstract  The article is concerned with the procedure of insertion of optimizable fragments of route solutions into the global solutions of the “big’’ problem defined by heuristic algorithms. Setting of the route problem takes into account some singularities of the engineering problem about the sequential cutting of details each having one exterior and probably several interior contours. The latter ones must be subjected to cutting previously in comparison with the exterior contour, which leads to a great number of given preceding conditions. These conditions are actively used to decrease the computational complexity. Nevertheless, the problem dimensionality remains sufficiently large that does not permit to use “global’’ dynamic programming and forces heuristic algorithms to be used (the problem under investigation is a hardsolvable problem in the traditional sense). Therefore, it is interesting to develop the methods for correction of solutions based on the abovementioned algorithms. In the present investigation, such correction is realized by the replacement of fragments (of the abovementioned solutions) having a moderate dimensionality by optimal “blocks’’ constructed by dynamic programming with local preceding conditions which are compatible with the constraints of the initial “big’’ problem. The proposed replacement does not deteriorate, but, in typical cases, improves the quality of the initial heuristic solution. This is verified by the computing experiment on multicore computer. The proposed algorithm is realized in the iterated regime: the solution (in the form of “routetrace’’) obtained after the first insertion on the basis of dynamic programming is taken as an initial solution for which the insertion is constructed again. In addition, the beginning of the new insertion is chosen randomly in the bounds defined by the possibilities of formation of a sliding “window’’ of the appreciable dimensionality which is in fact sufficient for the employment of the economical version of dynamic programming. Further, the procedure is repeated. The operation of the iterated algorithm is illustrated by solution of model problems including the versions with sufficiently dense “packing’’ of parts on a sheet, which is typical for the engineering production. 
Keywords  routing problem, preceding conditions 
UDC  519.6 
MSC  93CXX, 93С15 
DOI  10.20537/vm140204 
Received  1 March 2014 
Language  Russian 
Citation  Petunin A.A., Chentsov A.G., Chentsov P.A. Local dynamic programming incuts in routing problems with restrictions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 5675. 
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