Abstract
In this work, using the group theoretical approach we point out so me conditions on the B−1N matrix, often verified in practice, that make it possible to transform the system of linear congruences (constraints of problem 2) in a block diagonal form. In some cases, using this proce dure, the number of constraints can increase with respect to the number of constraints of problem 2. However, the problem can be solved indipen dently for the variables associated with each block.
This procedure leads to the indipendent solution of a number of subproblems in a smaller number of variables.
In the worst case each subproblem requires the same number of constraints as the original problem, but generally this number is smaller.
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Keywords
- Integer Programming
- Integer Linear Programming
- Linear Programming Problem
- Mixed Integer Linear Programming
- Decomposition Technique
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References
1 J.J.H. FORREST, J.P.H. HIRST, J.A. TOULIN, Practical solution of large mixed integer programming problems with UMPIRE. Management Science, vol. 20, n. 5, January 1974.
2 G. MITRA, Investigation of some branch and bound algorithms for (0–1) mixed integer linear programming. Mathematical Programming 4, pp. 155–170, 1973.
3 M. SHAW, Review of computational experience in solving large mixed integer programming problems. pp. 406–412. Applications of Mathematical Programming Techniques, English Universities Press, London 1970.
4 A.M. GEOFFRION, G.W. GRAVES, Multicommodity distribution system design by Benders decomposition. Management Science, vol. 20, n. 5, January 1974.
5 G. GALLO, E. MARTINO, B. SIMEONE, Group optimization algorithms and some numerical results via a branch and bound approach. Giornate AICA su "Tecniche di Simulazione ed Algoritmi". Mila no, Nov. 1972.
6 J.F. SHAPIRO, Dynamic programming algorithms for the integer programming problem I: the integer programming problem viewed as a knapsack-type problem. Operation Research, 16 January 1968.
7 J.F. SHAPIRO, Group theoretic algorithms for the integer programming problem II: extension to a general algorithm. Operation Research 16, September 1968.
8 J.A. TOMLIN, Branch and bound methods for integer and non-convex programming. Integer and non-linear programming, cap. 21, North-Holland, Amsterdam 1970.
9 L.A. WOLSEY, Extensions of the group theoretic approach in integer programming. Management Science, vol. 18, n. 1, September 1971.
10 S. ZIONTS, Linear and integer programming, Prentice-Hall, 1974.
11 T.C. HU, Integer programming and network follows. Addison-Wesley Publishing Company, 1969.
12 R.E. GOMORY, Some polyhedra related to combinatorial problems. Linear Algebra and Its Applications, n. 2, 1969.
13 D.E. BELL, Improved bounds for integer programs: a supergroup approach. Research Memorandum of IIASA, November 1973.
14 H. GREENBERG, Integer programming. Academic Press, 1971.
15 A.M. GEOFFRION, Lagrangean relaxtion and its uses in integer programming. Western Management Science Institute, Working Paper n. 195.
16 M.L. FISHER, J.F. SHAPIRO, Constructive duality in integer program ming. Massachusetts Institute of Technology. Working Paper OK 008-72, April 1972.
17 R.S. GARFINKEL, G.L. NEMHAUSER, Integer programming. John Wiley and Sons, 1972.
18 G.S. MOSTOW J.H SAMSON, I.P. MEYER, Fundamental structure of algebra. Mc Graw-Hill, New York, 1963.
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Giulianelli, S., Lucertini, M. (1976). A decomposition technique in integer linear programming. In: Cea, J. (eds) Optimization Techniques Modeling and Optimization in the Service of Man Part 2. Optimization Techniques 1975. Lecture Notes in Computer Science, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-07623-9_282
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