NP-completeness of cell formation problem with grouping efficacy objective

In the current paper we provide a proof of NP-completeness for the Cell Formation Problem (CFP) with the fractional grouping efficacy objective function. First the CFP with a linear objective function is considered. Following the ideas of Pinheiro et al. (2016) we show that it is equivalent to the Bicluster Graph Editing Problem (BGEP), which is known to be NP-complete due to the reduction from the 3-Exact 3-Cover Problem – 3E3CP (Amit, 2004). Then we suggest a polynomial reduction of the CFP with the linear objective to the CFP with the grouping efficacy objective. It proves the NP-completeness of this fractional CFP formulation. Along with the NP-status our paper presents important connections of the CFP with the BGEP and 3E3CP. Such connections could be used for ”transferring” of known theoretical properties, efficient algorithms, polynomial cases, and other features of well-studied graph editing and exact covering problems to the CFP.

[1]  John L. Burbidge,et al.  The introduction of group technology , 1975 .

[2]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[3]  Harold J. Steudel,et al.  A within-cell utilization based heuristic for designing cellular manufacturing systems , 1987 .

[4]  M. Chandrasekharan,et al.  Grouping efficacy: a quantitative criterion for goodness of block diagonal forms of binary matrices in group technology , 1990 .

[5]  Damodar Y. Golhar,et al.  The just-in-time philosophy : a literature review , 1991 .

[6]  Warren J. Boe,et al.  DESIGNING CELLULAR MANUFACTURING SYSTEMS: BRANCH-AND-BOUND AND A∗ APPROACHES , 1993 .

[7]  Angappa Gunasekaran,et al.  An investigation into the application of group technology in advanced manufacturing systems , 1994 .

[8]  Dana S. Nau,et al.  Manufacturing cell formation by state-space search , 1996, Ann. Oper. Res..

[9]  S. Sofianopoulou,et al.  An optimal tree search method for the manufacturing systems cell formation problem , 1998, Eur. J. Oper. Res..

[10]  Kai-Ling Mak,et al.  An Adaptive Genetic Algorithm for Manufacturing Cell Formation , 2000 .

[11]  Ali M. S. Zalzala,et al.  Recent developments in evolutionary computation for manufacturing optimization: problems, solutions, and comparisons , 2000, IEEE Trans. Evol. Comput..

[12]  Bopaya Bidanda,et al.  Worker assignment in cellular manufacturing considering technical and human skills , 2002 .

[13]  Mauricio G. C. Resende,et al.  An evolutionary algorithm for manufacturing cell formation , 2004, Comput. Ind. Eng..

[14]  Jayme Luiz Szwarcfiter,et al.  Applying Modular Decomposition to Parameterized Cluster Editing Problems , 2008, Theory of Computing Systems.

[15]  Tabitha L. James,et al.  A hybrid grouping genetic algorithm for the cell formation problem , 2007, Comput. Oper. Res..

[16]  Yong Zhang,et al.  Improved Algorithms for Bicluster Editing , 2008, TAMC.

[17]  Panos M. Pardalos,et al.  Biclustering in data mining , 2008, Comput. Oper. Res..

[18]  Felix T.S. Chan,et al.  Cell formation problem with consideration of both intracellular and intercellular movements , 2008 .

[19]  C. Hicks,et al.  An Enhanced Grouping Genetic Algorithm for solving the cell formation problem , 2009 .

[20]  Alexandre Dolgui,et al.  Supply Chain Engineering , 2010 .

[21]  Maghsud Solimanpur,et al.  Solving cell formation problem in cellular manufacturing using ant-colony-based optimization , 2010 .

[22]  Nir Ailon,et al.  Improved Approximation Algorithms for Bipartite Correlation Clustering , 2011, SIAM J. Comput..

[23]  Tai-Hsi Wu,et al.  An efficient tabu search algorithm to the cell formation problem with alternative routings and machine reliability considerations , 2011, Comput. Ind. Eng..

[24]  J. Arkat,et al.  A new branch and bound algorithm for cell formation problem , 2012 .

[25]  Jacques A. Ferland,et al.  An exact method for solving the manufacturing cell formation problem , 2012 .

[26]  Boris Goldengorin,et al.  An exact model for cell formation in group technology , 2012, Comput. Manag. Sci..

[27]  Mohammad Mahdi Paydar,et al.  A hybrid genetic-variable neighborhood search algorithm for the cell formation problem based on grouping efficacy , 2013, Comput. Oper. Res..

[28]  Gürsel A. Süer,et al.  Stochastic skill-based manpower allocation in a cellular manufacturing system , 2014 .

[29]  Panos M. Pardalos,et al.  Exact model for the cell formation problem , 2014, Optim. Lett..

[30]  Peng Sun,et al.  Complexity of Dense Bicluster Editing Problems , 2014, COCOON.

[31]  Peng Sun,et al.  Bi-Force: large-scale bicluster editing and its application to gene expression data biclustering , 2014, Nucleic acids research.

[32]  Somnath Sikdar,et al.  Fast Biclustering by Dual Parameterization , 2015, IPEC.

[33]  Youkyung Won,et al.  Effective two-phase p-median approach for the balanced cell formation in the design of cellular manufacturing system , 2015 .

[34]  Panos M. Pardalos,et al.  Pareto-optimal front of cell formation problem in group technology , 2014, Journal of Global Optimization.

[35]  M. Brusco.  An exact algorithm for maximizing grouping efficacy in part–machine clustering , 2015 .

[36]  Vangelis Th. Paschos,et al.  Parameterized Algorithms for the Max K-set Cover and Related Satisfiability Problems Parameterized Exact and Approximation Algorithms for Maximum K-set Cover and Related Satisfiability Problems , 2022 .

[37]  Farouq Alhourani.  Cellular manufacturing system design considering machines reliability and parts alternative process routings , 2016 .

[38]  Ivan C. Martins,et al.  On Solving Manufacturing Cell Formation via Bicluster Editing , 2013, Eur. J. Oper. Res..

[39]  Irina Utkina,et al.  A Branch and Bound Algorithm for a Fractional 0-1 Programming Problem , 2016, DOOR.

[40]  Joseph Y.-T. Leung,et al.  Solving cell formation and task scheduling in cellular manufacturing system by discrete bacteria foraging algorithm , 2016 .

[41]  Reza Tavakkoli-Moghaddam,et al.  Towards a psychologically consistent cellular manufacturing system , 2017 .

[42]  Alexandre Dolgui,et al.  Decision support for design of reconfigurable rotary machining systems for family part production , 2017 .

[43]  Sebastián Lozano,et al.  Cell design and multi-period machine loading in cellular reconfigurable manufacturing systems with alternative routing , 2017, Int. J. Prod. Res..

[44]  Fábio Protti,et al.  New heuristics for the Bicluster Editing Problem , 2017, Ann. Oper. Res..

[45]  Irina Utkina,et al.  A branch-and-bound algorithm for the cell formation problem , 2018, Int. J. Prod. Res..

[46]  Manoj Kumar Tiwari,et al.  Design for manufacturing and assembly/disassembly: joint design of products and production systems , 2018, Int. J. Prod. Res..

[47]  Yiyo Kuo,et al.  Operator assignment with cell loading and product sequencing in labour-intensive assembly cells – a case study of a bicycle assembly company , 2018 .

[48]  Mikhail Batsyn,et al.  An efficient exact model for the cell formation problem with a variable number of production cells , 2017, Comput. Oper. Res..

[49]  Chengkuan Zeng,et al.  Auction-based cooperation mechanism for cell part scheduling with transportation capacity constraint , 2018, Int. J. Prod. Res..