# A Path Relinking Algorithm for the Generalized Assignment Problem

## Abstract

The Generalized Assignment Problem (GAP) consists in finding a maximum-profit assignment of tasks to agents with capacity constraints. In this paper, a path relinking heuristic is proposed for the GAP. The main feature of our path relinking is that both feasible and infeasible solutions are inserted in the reference set of elite solutions, trade-off between feasibility and infeasibility being ruled through a penalty coefficient for infeasibility. Since exploration of the solution space is very sensitive to this penalty coefficient, the coefficient is dynamically updated at each round of combinations so that a balance is kept between feasible and infeasible solutions in the reference set. Numerical experiments reported on classical testbed instances of the OR-library show that the algorithm compares favorably to several other methods in the literature. In particular, more than 95% of the instances in the test-file were solved to optimality with short computation time.

## Keywords

Combinatorial optimization Generalized assignment Metaheuristics Path relinking.## Preview

Unable to display preview. Download preview PDF.

## Bibliography

- M.M. Amini and M. Racer. A rigorous comparison of alternative solution methods for the generalized assignment problem.
*Management Sc*, 40: 868890, 1994.Google Scholar - P. Barcia and K. Jornsten. Improved lagrangian decomposition: An application to the generalized assignment problem.
*European J. of Oper. Res*, 46: 84–92, 1990.zbMATHCrossRefGoogle Scholar - D.G. Cattrysse, M. Salomon, and L.N. van Wassenhove. A set partitioning heuristic for the generalized assignment problem.
*European J. of Oper. Res*, 72: 167–174, 1994.zbMATHCrossRefGoogle Scholar - D.G. Cattrysse and L.N. van Wassenhove. A survey of algorithms for the gen-eralized assignment problem.
*European J. of Oper. Res*, 60: 260–272, 1992.Google Scholar - P.C. Chu and J.E. Beasley. A genetic algorithm for the generalised assignment problem.
*Comp. Oper. Res*, 24: 17–23, 1997.MathSciNetzbMATHCrossRefGoogle Scholar - J. Diaz and E. Fernandez. A tabu search heuristic for the generalized assignment problem.
*European J. of Oper. Res*, 132: 1: 22–38, 2001.MathSciNetzbMATHCrossRefGoogle Scholar - M.L. Fisher, R. Jaikumar, and L.N. Van Wassenhove. A multiplier adjustment method for the generalized assignment problem.
*Management Sc*, 32: 9: 10951103, 1986.Google Scholar - F. Glover. Genetic algorithms, evolutionary algorithms and scatter search: Changing tides and untapped potentials.
*INFORMS Computer Science Newsletter*, 19: 1: 7–14, 1998a.Google Scholar - F. Glover. A template for scatter search and path relinking. In J.-K. Hao, E. Lut-ton, E. Ronald, M. Schoenauer, and D. Snyers, editors,
*Artificial Evolution*,*Lecture Notes in Computer Science*, pages 13–54. Springer, 1998b.Google Scholar - F. Glover, J.P. Kelly, and M. Laguna. Genetic algorithms and tabu search: Hybrids for optimization.
*Computers and Oper. Res*, 22: 1: 111–134, 1994.CrossRefGoogle Scholar - M. Guignard and M. Rosenwein. An improved dual-based algorithm for the generalized assignment problem.
*Oper. Res*, 37: 4: 658–663, 1989.MathSciNetzbMATHCrossRefGoogle Scholar - A.J. Higgins. A dynamic tabu search for large-scale generalised assignment problems.
*Computers and Oper. Res*, 28: 10: 1039–1048, 2001.MathSciNetzbMATHCrossRefGoogle Scholar - M. Laguna, J.P. Kelly, J.L. Gonzalez-Velarde, and F. Glover. Tabu search for the multilevel generalized assignment problem.
*European J. of Oper. Res*, 82: 176–189, 1995.zbMATHCrossRefGoogle Scholar - M. Laguna, R. Marti, and V. Campos. Intensification and diversification with elite tabu search solutions for the linear ordering problem.
*Computers and OR*, 26: 1217–1230, 1999.zbMATHCrossRefGoogle Scholar - L.A.N. Lorena and M.G. Narciso. Relaxation heuristics for a generalized assignment problem.
*European J. of Oper. Res*, 91: 600–610, 1996.zbMATHCrossRefGoogle Scholar - S. Martello and P. Toth. An algorithm for the generalized assignment problem. In J.P. Brans, editor,
*Oper. Res*. ‘81, pages 589–603. North Holland, 1981.Google Scholar - I.H. Osman. Heuristics for the generalized assignment problem: Simulated annealing and tabu search approaches.
*OR Spektrum*, 17: 211–225, 1995.zbMATHCrossRefGoogle Scholar - A. Plateau, D. Tachat, and P. Tolla. A hybrid search combining interior point methods and metaheuristics for 0–1 programming.
*Intl. Trans. in Oper. Res*, 9: 6: 731–746, 2002.MathSciNetzbMATHCrossRefGoogle Scholar - G.T. Ross and R.M. Soland. A branch and bound algorithm for the generalized assignment problem.
*Math. Prog*, 8: 91–103, 1975.MathSciNetzbMATHCrossRefGoogle Scholar - M. Savelsbergh. A branch and cut algorithm for the generalized assignment problem.
*Oper. Res*, 45: 6: 831–841, 1997.MathSciNetzbMATHCrossRefGoogle Scholar - M. Trick. A linear relaxation heuristic for the generalized assignment problem.
*Naval Research Logistics*, 39: 137–151, 1992.MathSciNetzbMATHCrossRefGoogle Scholar - J.M. Wilson. A genetic algorithm for the generalised assignment problem.
*J. of the Oper. Res. Soc*, 48: 804–809, 1997.zbMATHGoogle Scholar - M. Yagiura, T. Ibaraki, and F. Glover. An ejection chain approach for the generalized assignment problem. Technical Report 99013, Department of Applied Mathematics and Physics, Graduate Sch. of Informatics, Kyoto University, 1999.Google Scholar
- M. Yagiura, T. Ibaraki., and F. Glover. A path relinking approach for the generalized assignment problem. In
*Proc. International Symposium on Scheduling*,*Japan*,*June 4–6*,pages 105–108, 2002. To appear in*INFORMS J. on Computing*with title “An Ejection Chain Approach for the Generalized Assignment Problem.”.Google Scholar *M. Yagiura, T. Yamaguchi, and T. Ibaraki. A variable depth search algorithm with branching search for the generalized assignment problem*Optimization Methods and Software*10:419–441, 1998*Google Scholar