Optimal algorithm for the demand routing problem in multicommodity flow distribution networks with diversification constraints and concave costs

Distribution problems are of high relevance within the supply chain system. In real life situations various different commodities may flow in the distribution process. Furthermore, the connection between production and demand centres makes use of complex mesh networks that can include diversification constraints to avoid overcharged paths. In addition, the consideration in certain situations of economies of scale gives rise to non-linear cost functions that make it difficult to deal with an optimal routing scheme. This problem is well represented by the multicommodity flow distribution networks with diversification constraints and concave costs (MFDCC) problem. Here we present an optimal algorithm based on the Kuhn–Tucker optimality conditions of the problem and capable of supplying optimal distribution routes in such complex networks. The algorithm follows an iterative procedure. Each iteration constructive solutions are checked with respect to the Kuhn–Tucker optimality conditions. Solutions consider a set of paths transporting all the demand allowed by its diversification constraint (saturated paths), a set of empty paths, and an indicator path transporting the remaining demand to satisfy the demand equation. The algorithm reduces the total cost in the network in a monotonic sequence to the optimum. The algorithm was tested in a trial library and the optimum was reached for all the instances. The algorithm showed a major dependency with respect to the number of nodes and arcs of the graph, as well as the density of arcs in the graph.

[1]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[2]  Pablo Cortés,et al.  Andalucía Assesses the Investment Needed to Deploy a Fiber-Optic Network , 2006, Interfaces.

[3]  Alice E. Smith,et al.  Local search genetic algorithm for optimal design of reliable networks , 1997, IEEE Trans. Evol. Comput..

[4]  F. Arcelus,et al.  Green logistics at Eroski: A case study , 2011 .

[5]  Shangyao Yan,et al.  Probabilistic local search algorithms for concave cost transportation network problems , 1999, Eur. J. Oper. Res..

[6]  G. D. Smith,et al.  Solving the Graphical Steiner Tree Problem Using Genetic Algorithms , 1993 .

[7]  Phan Thien Thach.  A decomposition method using a pricing mechanism for min concave cost flow problems with a hierarchical structure , 1992, Math. Program..

[8]  Yu-Chung Tsao,et al.  A MULTI-ITEM SUPPLY CHAIN WITH CREDIT PERIODS AND WEIGHT FREIGHT COST DISCOUNTS , 2012 .

[9]  F. L. Hitchcock.  The Distribution of a Product from Several Sources to Numerous Localities , 1941 .

[10]  W. Zangwill.  Minimum Concave Cost Flows in Certain Networks , 1968 .

[11]  Ismail Karaoglan,et al.  An adaptive tabu-simulated annealing for concave cost transportation problems , 2008, J. Oper. Res. Soc..

[12]  Markku Tuominen,et al.  The development of a decision support system in multimodal transportation routing within Greater Mekong sub-region countries , 2012 .

[13]  Gengui Zhou,et al.  A genetic algorithm approach on tree-like telecommunication network design problem , 2003, J. Oper. Res. Soc..

[14]  M. A. Hanson,et al.  Necessary and sufficient conditions in constrained optimization , 1987, Math. Program..

[15]  Rainer E. Burkard,et al.  Linear Approximations in a Dynamic Programming Approach for the Uncapacitated Single-Source Minimum Concave Cost Network Flow Problem in Acyclic Networks , 2001, J. Glob. Optim..

[16]  Jon M. Kleinberg,et al.  Approximation algorithms for disjoint paths problems , 1996 .

[17]  Alice E. Smith,et al.  Optimal Design of Reliable Computer Networks: A Comparison of Metaheuristics , 2003, J. Heuristics.

[18]  Mikael Rönnqvist,et al.  A Lagrangean heuristic for the capacitated concave minimum cost network flow problem , 1994 .

[19]  M. Florian,et al.  DETERMINISTIC PRODUCTION PLANNING WITH CONCAVE COSTS AND CAPACITY CONSTRAINTS. , 1971 .

[20]  Pablo Cortés,et al.  Genetic algorithm for planning cable telecommunication networks , 2001, Appl. Soft Comput..

[21]  Larry J. LeBlanc.  Global Solutions for a Nonconvex Nonconcave Rail Network Model , 1976 .

[22]  Mordecai Avriel,et al.  Nonlinear programming , 1976 .