Nonlinear fixed charge transportation problem by minimum cost flow-based genetic algorithm

Nonlinear Fixed Charge Transportation Problem (NFCTP) is a variant of fixed charge transportation problem, which is to ship available amounts of goods to satisfy the demands at minimal total cost, on condition that any route has a fixed cost irrelative to its shipping amount if it is used, and a variable cost directly proportional to the quadratic of its shipping amount as a nonlinear term. This paper aims at developing an efficient method to solve NFCTP. In this paper, NFCTP is formulated using a mixed integer programming model. Based on steady-state genetic algorithm as framework, and minimum cost flow algorithm as decoder, a hybrid genetic algorithm named NFCTP-HGA is proposed as a solution method of the model. Taking advantage of nonlinear structure and special network structure of NFCTP, the NFCTP-HGA algorithm has good performance in the sense of being implemented on computer, computational time, required memory for computation, and ability to search global optimum. The application of the NFCTP-HGA algorithm is illustrated with examples. Numerical experiments demonstrate that the NFCTP-HGA algorithm is an efficient and robust method to solve NFCTP, especially applicable to large scale problems.

[1]  Mitsuo Gen,et al.  Genetic algorithms and engineering optimization , 1999 .

[2]  Mitsuo Gen,et al.  Spanning tree-based genetic algorithm for the bicriteria fixed charge transportation problem , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[3]  Béla Bollobás,et al.  Modern Graph Theory , 2002, Graduate Texts in Mathematics.

[4]  N. Jawahar,et al.  A genetic algorithm for the two-stage supply chain distribution problem associated with a fixed charge , 2009, Eur. J. Oper. Res..

[5]  Yale T. Herer,et al.  Fast Algorithms for Single-Sink Fixed Charge Transportation Problems with Applications to Manufacturing and Transportation , 1996, Transp. Sci..

[6]  Jean-Marie Proth,et al.  Production , Manufacturing and Logistics The concave cost supply problem , 2003 .

[7]  W. M. Hirsch,et al.  The fixed charge problem , 1968 .

[8]  Lixing Yang,et al.  Fuzzy fixed charge solid transportation problem and algorithm , 2007, Appl. Soft Comput..

[9]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[10]  Chandrasekharan Rajendran,et al.  A genetic algorithm for solving the fixed-charge transportation model: Two-stage problem , 2012, Comput. Oper. Res..

[11]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[12]  Keith Sandrock,et al.  A Simple Algorithm for Solving Small, Fixed-Charge Transportation Problems , 1988 .

[13]  Patrick G. McKeown,et al.  An algorithm for solving fixed-charge problems using surrogate constraints , 1991, Comput. Oper. Res..

[14]  M. W. Cooper,et al.  A branch and bound method for the fixed charge transportation problem , 1970 .

[15]  K. Dejong,et al.  An Analysis Of The Behavior Of A Class Of Genetic Adaptive Systems , 1975 .

[16]  Daniel E. O'Leary,et al.  Use of penalties in a branch and bound procedure for the fixed charge transportation problem , 1989 .

[17]  Benjamin Lev,et al.  Modeling fixed-charge problems with polynomials , 2011 .

[18]  Andreas Klose,et al.  Algorithms for solving the single-sink fixed-charge transportation problem , 2008, Comput. Oper. Res..

[19]  Katta G. Murty,et al.  Solving the Fixed Charge Problem by Ranking the Extreme Points , 1968, Oper. Res..

[20]  Jerzy Kamburowski,et al.  On the minimum cost project schedule , 1995 .

[21]  P. McKeown.  A branch‐and‐bound algorithm for solving fixed charge problems , 1981 .

[22]  Veena Adlakha,et al.  A heuristic method for 'more-for-less' in distribution-related problems , 2001 .

[23]  Anton V. Eremeev,et al.  Genetic algorithms for a supply management problem: MIP-recombination vs greedy decoder , 2009, Eur. J. Oper. Res..

[24]  Panos M. Pardalos,et al.  A solution approach to the fixed charge network flow problem using a dynamic slope scaling procedure , 1999, Oper. Res. Lett..

[25]  Veena Adlakha,et al.  On the fixed-charge transportation problem , 1999 .

[26]  Fanrong Xie,et al.  A note on "Nonlinear fixed charge transportation problem by spanning tree-based genetic algorithm" by Jung-Bok Jo, Yinzhen Li, Mitsuo Gen, Computers & Industrial Engineering (2007) , 2010, Comput. Ind. Eng..

[27]  A. Victor Cabot,et al.  Some branch‐and‐bound procedures for fixed‐cost transportation problems , 1984 .

[28]  Minghe Sun,et al.  A tabu search heuristic procedure for the fixed charge transportation problem , 1998, Eur. J. Oper. Res..

[29]  Ulrich Brenner,et al.  A faster polynomial algorithm for the unbalanced Hitchcock transportation problem , 2008, Oper. Res. Lett..

[30]  Moustapha Diaby,et al.  Successive Linear Approximation Procedure for Generalized Fixed-Charge Transportation Problems , 1991 .

[31]  Bruce W. Lamar,et al.  Revised-Modified Penalties for Fixed Charge Transportation Problems , 1997 .

[32]  Patrick G. McKeown,et al.  The pure fixed charge transportation problem , 1979 .

[33]  M. Balinski.  Fixed‐cost transportation problems , 1961 .

[34]  Chandrasekharan Rajendran,et al.  Fast heuristic algorithms to solve a single-stage Fixed-Charge Transportation Problem , 2009 .

[35]  Benjamin Lev,et al.  More-for-less algorithm for fixed-charge transportation problems , 2007 .

[36]  Colin R. Reeves,et al.  Genetic Algorithms for the Operations Researcher , 1997, INFORMS J. Comput..

[37]  Herminia I. Calvete.  Network simplex algorithm for the general equal flow problem , 2003, Eur. J. Oper. Res..

[38]  Satoru Fujishige.  A maximum flow algorithm using MA ordering , 2003, Oper. Res. Lett..

[39]  K. Spielberg.  On the fixed charge transportation problem , 1964, ACM National Conference.

[40]  Reza Tavakkoli-Moghaddam,et al.  Addressing a nonlinear fixed-charge transportation problem using a spanning tree-based genetic algorithm , 2010, Comput. Ind. Eng..

[41]  Paul Gray,et al.  Technical Note - Exact Solution of the Fixed-Charge Transportation Problem , 1971, Oper. Res..

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

[43]  Rasajit Kumar Bera,et al.  A modified approach to the large amplitude free vibration of parabolic plates , 1995 .

[44]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[45]  N. Jawahar,et al.  A Simulated Annealing Algorithm for a two-stage fixed charge distribution problem of a Supply Chain , 2010 .

[46]  H. Arsham,et al.  A Simplex-Type Algorithm for General Transportation Problems: An Alternative to Stepping-Stone , 1989 .

[47]  Z. Michalewicz,et al.  A genetic algorithm for the linear transportation problem , 1991, IEEE Trans. Syst. Man Cybern..

[48]  H. D. Ratliff,et al.  A branch-and-bound method for the fixed charge transportation problem , 1990 .

[49]  Abraham Charnes,et al.  The Stepping Stone Method of Explaining Linear Programming Calculations in Transportation Problems , 1954 .

[50]  Fanrong Xie,et al.  A Heuristic Algorithm for Solving Fixed-Charge Transportation Problem , 2009, 2009 WRI World Congress on Computer Science and Information Engineering.

[51]  Veena Adlakha,et al.  A quick sufficient solution to the more-for-less paradox in the transportation problem , 1998 .

[52]  Bernard W. Taylor,et al.  Operations Management: Focusing on Quality and Competitiveness , 1999 .

[53]  Norman J. Driebeek.  An Algorithm for the Solution of Mixed Integer Programming Problems , 1966 .

[54]  Kenneth Alan De Jong,et al.  An analysis of the behavior of a class of genetic adaptive systems. , 1975 .

[55]  Mitsuo Gen,et al.  Recent network design techniques using evolutionary algorithms , 2005 .

[56]  H. Arsham,et al.  Postoptimality Analyses of the Transportation Problem , 1992 .

[57]  Panos M. Pardalos,et al.  Cross-facility management of production and transportation planning problem , 2006, Comput. Oper. Res..

[58]  Guangxing Zeng,et al.  On Minimum Duration Project Schedule , 2010, 4th International Conference on New Trends in Information Science and Service Science.

[59]  Veena Adlakha,et al.  A note on the procedure MFL for a more-for-less solution in transportation problems , 2000 .

[60]  Fred W. Glover,et al.  A New Optimization Method for Large Scale Fixed Charge Transportation Problems , 1981, Oper. Res..

[61]  V. E. Unger,et al.  The Group-Theoretic Structure in the Fixed-Charge Transportation Problem , 1973, Oper. Res..

[62]  G. M. Kannan,et al.  Comments on Nonlinear fixed charge transportation problem by spanning tree-based genetic algorithm by Jung-Bok Jo, Yinzhen Li, Mitsuo Gen, Computers & Industrial Engineering (2007) , 2008 .

[63]  Shiang-Tai Liu,et al.  The total cost bounds of the transportation problem with varying demand and supply , 2003 .

[64]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[65]  Benjamin Lev,et al.  A branching method for the fixed charge transportation problem , 2010 .

[66]  Mitsuo Gen,et al.  Nonlinear fixed charge transportation problem by spanning tree-based genetic algorithm , 2007, Comput. Ind. Eng..

[67]  Kalpana Dahiya,et al.  Capacitated transportation problem with bounds on RIM conditions , 2007, Eur. J. Oper. Res..

[68]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[69]  Reza Tavakkoli-Moghaddam,et al.  Solving a capacitated fixed-charge transportation problem by artificial immune and genetic algorithms with a Prüfer number representation , 2011, Expert Syst. Appl..

[70]  Bruce W. Lamar,et al.  Capacity improvement, penalties, and the fixed charge transportation problem , 1999 .

[71]  Veena Adlakha,et al.  A SIMPLE HEURISTIC FOR SOLVING SMALL FIXED-CHARGE TRANSPORTATION PROBLEMS , 2003 .