A Branch and Bound Algorithm for a Fractional 0-1 Programming Problem

We consider a fractional 0-1 programming problem arising in manufacturing. The problem consists in clustering of machines together with parts processed on these machines into manufacturing cells so that intra-cell processing of parts is maximized and inter-cell movement is minimized. This problem is called Cell Formation Problem (CFP) and it is an NP-hard optimization problem with Boolean variables and constraints and with a fractional objective function. Because of its high computational complexity there are a lot of heuristics developed for it. In this paper we suggest a branch and bound algorithm which provides exact solutions for the CFP with a variable number of cells and grouping efficacy objective function. This algorithm finds optimal solutions for 21 of the 35 popular benchmark instances from literature and for the remaining 14 instances it finds good solutions close to the best known.

[1]  A. Kusiak.  The generalized group technology concept , 1987 .

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

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

[4]  M. Chandrasekharan,et al.  GROUPABIL1TY: an analysis of the properties of binary data matrices for group technology , 1989 .

[5]  J. King.  Machine-component grouping in production flow analysis: an approach using a rank order clustering algorithm , 1980 .

[6]  Larry R. Taube,et al.  Weighted similarity measure heuristics for the group technology machine clustering problem , 1985 .

[7]  Allan S. Carrie,et al.  Numerical taxonomy applied to group technology and plant layout , 1973 .

[8]  Hamid Seifoddini,et al.  A note on the similarity coefficient method and the problem of improper machine assignment in group technology applications , 1989 .

[9]  Warren J. Boe,et al.  A close neighbour algorithm for designing cellular manufacturing systems , 1991 .

[10]  Paul J. Schweitzer,et al.  Problem Decomposition and Data Reorganization by a Clustering Technique , 1972, Oper. Res..

[11]  J. King,et al.  Machine-component group formation in group technology: review and extension , 1982 .

[12]  Panos M. Pardalos,et al.  Heuristic Algorithm for the Cell Formation Problem , 2013 .

[13]  Anthony Vannelli,et al.  Strategic subcontracting for efficient disaggregated manufacturing , 1986 .

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

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

[16]  F. Boctor.  A Jinear formulation of the machine-part cell formation problem , 1991 .

[17]  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..

[18]  Ronald G. Asktn,et al.  A cost-based heuristic for group technology configuration† , 1987 .

[19]  Larry E. Stanfel,et al.  Machine clustering for economic production , 1985 .

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

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

[22]  D. A. Milner,et al.  Direct clustering algorithm for group formation in cellular manufacture , 1982 .

[23]  Andrew Kusiak,et al.  Grouping of parts and components in flexible manufacturing systems , 1986 .

[24]  Philip M. Wolfe,et al.  Application of the Similarity Coefficient Method in Group Technology , 1986 .

[25]  M. Chandrasekharan,et al.  ZODIAC—an algorithm for concurrent formation of part-families and machine-cells , 1987 .

[26]  Geetha Srinivasan,et al.  A heuristic algorithm for the cell formation problem , 1994 .

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

[28]  M. Chandrasekharan,et al.  An ideal seed non-hierarchical clustering algorithm for cellular manufacturing , 1986 .

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

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

[31]  T. Narendran,et al.  An assignment model for the part-families problem in group technology , 1990 .

[32]  Jacques A. Ferland,et al.  Cell Formation Problem Solved Exactly with the Dinkelbach Algorithm Bouazza Elbenani , 2012 .

[33]  Panos M. Pardalos,et al.  Models, algorithms, and technologies for network analysis , 2013 .

[34]  Panos M. Pardalos,et al.  Cell Formation in Industrial Engineering: Theory, Algorithms and Experiments , 2013 .

[35]  P. Waghodekar,et al.  Machine-component cell formation in group technology: MACE , 1984 .

[36]  Larry R. Taube,et al.  The facets of group technology and their impacts on implementation--A state-of-the-art survey , 1985 .

[37]  M. Chandrasekharan,et al.  MODROC: an extension of rank order clustering for group technology , 1986 .

[38]  A. Kusiak,et al.  Efficient solving of the group technology problem , 1987 .

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

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

[41]  John L. Burbidge,et al.  The "new approach" to production , 1961 .