### Order Allocation for Stock Cutting in the Paper Industry

A common problem encountered in paper-production facilities is that of allocating customer orders to machines so as to minimize the total cost of production. It can be formulated as adual-angular integer program, with identical machines inducing symmetry. While the potential advantages of decomposing large mathematical programs into smaller subproblems have long been recognized, the solution of decomposableinteger programs remains extremely difficult. Symmetry intensifies the difficulty. This paper develops an approach, based on the construction of tight subproblem bounds, to solve decomposable dual-angular integer programs and successfully applies it to solve the problem from the paper industry. This method is of particular interest as it significantly reduces the impact of symmetry.

[1]  Odile Marcotte The cutting stock problem and integer rounding , 1985, Math. Program..

[2]  Martin W. P. Savelsbergh,et al.  A Branch-and-Price Algorithm for the Generalized Assignment Problem , 1997, Oper. Res..

[3]  Arthur M. Geoffrion,et al.  An Improved Implicit Enumeration Approach for Integer Programming , 1969, Oper. Res..

[4]  Moshe Ben-Horim,et al.  A linear programming approach , 1977 .

[5]  Pamela H. Vance,et al.  Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problem , 1998, Comput. Optim. Appl..

[6]  Linus Schrage Using decomposition in integer programming , 1973 .

[7]  George L. Nemhauser,et al.  Airline Crew Scheduling: A New Formulation and Decomposition Algorithm , 1997, Oper. Res..

[8]  Martin W. P. Savelsbergh,et al.  Branch-and-Price: Column Generation for Solving Huge Integer Programs , 1998, Oper. Res..

[9]  Dennis J. Sweeney,et al.  A Method of Decomposition for Integer Programs , 1979, Oper. Res..

[10]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

[11]  Ralph E. Gomory,et al.  A Linear Programming Approach to the Cutting Stock Problem---Part II , 1963 .

[12]  Harald Dyckhoff,et al.  A typology of cutting and packing problems , 1990 .

[13]  José M. Valério de Carvalho A Note on Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problems , 2002, Comput. Optim. Appl..

[14]  R. Gomory,et al.  A Linear Programming Approach to the Cutting-Stock Problem , 1961 .