Skip to main content
SpringerLink
Log in

The Multidimensional 0-1 Knapsack Problem—Bounds and Computational Aspects

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The multidimensional 0-1 knapsack problem (MKP) is a resource allocation model that is one of the most well-known integer programming problems. During the last few decades, an impressive amount of research on the 0-1 knapsack problem has been published in the literature, and efficient special-purpose methods have become available for solving very large-scale instances. However, the multidimensional case has received less attention from the operational research community. Although recent advances have made solving medium size instances possible, solving the NP-hard problem remains a very interesting challenge, especially when the number of constraints increases. This paper surveys the principal results published in the literature concerning both the problem's theoretical properties and its approximate or exact solutions. The paper focuses on the more recent results—for example, those relevant to surrogate and composite duality, new preprocessing approaches creating enhanced versions of leading commercial software, and efficient metaheuristic-based methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aboudi, R., A. Hallefjord, R. Helming, and K. Jornsten. (1989). “A Note on the Pivot and Complement Heuristic for 0-1 Programming Problems.” Operation Research Letters 8, 21–23.

    Article  Google Scholar 

  • Aboudi, R. and K. Jornsten. (1994). “Tabu Search for General Zero-One Integer Programs Using the Pivot and Complement Heuristics.” ORSA Journal of Computing 6, 82–93.

    Google Scholar 

  • Andonov, R., Balev S., A. Fréville, and N. Yanev. (2001). “A Dynamic Programming Based Reduction Procedure for the Multidimensional 0-1 Knapsack Problem.” Working paper, University of Valenciennes, presented at FRANCORO III, Québec, Canada, may 2001.

  • Averbakh, I. (1994). “Probabilistic Properties of the Dual Structure of the Multidimensional Knapsack Problem and Fast Statistically Efficient Algorithms.” Mathematical Programming 65, 311–330.

    Article  Google Scholar 

  • Balas, E., S. Ceria, M. Dawande, F. Margot, and G. Pataki. (2001). “OCTANE: A New Heuristic for Pure 0-1 Programs.” Operations Research 49, 207–225.

    Article  Google Scholar 

  • Balas, E. and C.H. Martin. (1980). “Pivot and Complement—A Heuristic for 0-1 Programming.” Management Sciences 26, 86–96.

    Google Scholar 

  • Balas, E. and E. Zemel. (1980). “An Algorithm for Large Zero-One Knapsack Problems.” Operations Research 28, 1130–1145.

    Google Scholar 

  • Battiti, R. and G. Tecchiolli. (1992). “Parallel Biased Search for Combinatorial Optimization: Genetic Algorithms and Tabu Search.” Microprocessors and Microsystems 16, 351–367.

    Article  Google Scholar 

  • Battiti, R. and G. Tecchiolli. (1994). “The Reactive Tabu Search.” ORSA Journal on Computing 6, 126– 140.

    Google Scholar 

  • Battiti, R. and G. Tecchiolli. (1995). “Local Search with Memory: Benchmarking RTS.” OR-Spektrum 17, 67–86.

    Article  Google Scholar 

  • Beasley, J.E. (1990). “OR-Library: Distributed Test Problems by Electronic Mail.” Journal of the Operational Research Society 41, 1069–1072 (http://mscmga.ms.ic.ac.uk/info.html).

  • Beaujon, G.J., S.P. Martin, and C.C. McDonald. (2001). “Balancing and Optimizing a Portfolio of R&D Projects.” Naval Research Logistics 48, 18–40.

    Article  Google Scholar 

  • Bixby, R.E., M. Fenelon, Z. Gu, E. Rothberg, and R. Wunderling. (2000). “MIP: Theory and Practice—Closing the Gap.” Research paper, ILOG CPLEX Division.

  • Bradley, G.H., P.L. Hammer, and L. Wolsey. (1974). “Coefficient Reduction for Inequalities in 0-1 Variables.” Mathematical Programming 7, 263–282.

    Article  Google Scholar 

  • Cabot, A.V. (1970). “An Enumeration Algorithm for Knapsack Problems.” Operations Research 18, 306–311.

    Google Scholar 

  • Chandra A.K., D.S. Hirschberg, and C.K. Wong. (1976). “Approximation Algorithms for Some Generalized Knapsack Problems.” Theoretical Computer Sciences 3, 293–304.

    Article  Google Scholar 

  • Chu, P. and J. Beasley. (1998). “A Genetic Algorithm for the Multidimensional Knapsack Problem.” Journal of Heuristics 4, 63–86.

    Article  Google Scholar 

  • CPLEX Optimization, Inc. (1994). “Using the CPLEX Callable Library and CPLEX Mixed Integer Library.” Version 3.0.

  • Crama, Y. and J. Mazzola. (1994). “On the Strength of Relaxations of Multidimensional Knapsack Problems.” INFOR 32, 219–225.

    Google Scholar 

  • Crowder, H., E. Johnson, and M. Padberg. (1983). “Solving Large-Scale Zero-One Linear Programming Problems.” Operations Research 31, 803–834.

    Article  Google Scholar 

  • Dammeyer, F. and S. Voss. (1991).“Application of Tabu Search Strategies for Solving Multiconstraint Zero-One Knapsack Problems.” Working paper, Technische Hochschule Darmstadt, Germany.

  • Dammeyer, F. and S. Voss. (1993). “Dynamic Tabu List Management Using Reverse Elimination Method.” Annals of Operations Research 31–46.

  • DASH Associates. (1994). XPRESS-MP User Manual.

  • Dietrich, B.L. and L.F. Escudero. (1990). “Coefficient Reduction Procedure for Knapsack-Like Constraints in 0-1 Programs with Variable Upper Bounds.” Operations Research Letters 9, 9–14.

    Article  Google Scholar 

  • Dietrich, B.L., L.F. Escudero, and F. Chance. (1993). “Effect Reformulation for 0-1 Programs: Methods and Computational Results.” Discrete Applied Mathematics 42, 147–175.

    Article  Google Scholar 

  • Dobson, G. (1982). “Worst-Case Analysis of Greedy Heuristics for Integer Programming with Non-Negative Date.” Mathematics of Operations Research 7, 515–531.

    Google Scholar 

  • Drexl, A. (1988). “A Simulated Annealing Approach to the Multiconstraint Zero-One Knapsack Problem.” Computing 40, 1–8.

    Article  Google Scholar 

  • Dueck, G. and T. Scheuer. (1990). “Threshold Accepting: A General Purpose Optimization Algorithm.” Journal of Computational Physics 90, 161–175.

    Article  Google Scholar 

  • Dueck, G. and J. Wirshing. (1989). “Threshold Accepting Algorithms for Multi-Constraint 0-1 Knapsack Problems.” Working paper, TR 89 10 016, IBM Heidelberg Scientific Center, Germany.

  • Dyer, H.E. (1980). “Calculating Surrogate Constraints.” Mathematical Programming 19, 255–278.

    Article  Google Scholar 

  • Dyer, M.E. and A.M. Frieze. (1989). “Probabilistic Analysis of the Multidimensional Knapsack Problem.” Mathematics of Operations Research 14, 162–176.

    Google Scholar 

  • Escudero, L.F., S. Martello, and P. Toth. (1996). “A Framework for Tightening 0-1 Programs Based on Extensions of Pure 0-1 KP and SS Problems.” In E. Balas and J. Clausen, (eds.), Proceedings of the 4th international IPCO Conference, Berlin: Springer, pp. 110–123.

  • Fayard, D. and G. Plateau. (1977). “Reduction Algorithm for Single and Multiple Constraints 0-1 Linear Programming Problems.” In Proceedings of Congress Methods of Mathematical Programming, Zakopane, Poland.

  • Fayard, D. and G. Plateau. (1982). “Algorithm 47: An Algorithm for the Solution of the Knapsack Problem.” Computing 28, 269–287.

    Article  Google Scholar 

  • Fischer, M.L. (1980). “Worst-Case Analysis of Heuristic Algorithms.” Management Sciences 26, 1–17.

    Google Scholar 

  • Fischer, M.L. (1981). “The Lagrangean Relaxation Method for Solving Integer Programming Problems.” Management Sciences 27, 1–18.

    Google Scholar 

  • Fontarani, J.F. (1995). “A Statistical Analysis of the Knapsack Problem.” Journal of Physics A—Mathematical and General 28, 4751–4759.

    Article  Google Scholar 

  • Fontarani, J.F. and R. Meir. (1993). “The Statistical Mechanics of the Ising Perceptron.” Journal of Physics A—Mathematical and General 26, 1077–1089.

    Article  Google Scholar 

  • Fréville, A. (2003). “The Multidimensional 0-1 Knapsack Problem: An Overview.” Invited Paper to Appear in European Journal of Operational Research.

  • Fréville, A. and S. Hanafi (2001). “Bornes duales robustes pour le sac à dos bi-dimensionnel en variables 0-1.” Working paper presented at FRANCORO III, Québec, Canada, May 2001.

  • Fréville, A. and F. Malca (2002). “Comparaison de méthodes pour la résolution du dual agrégé du sac à dos multidimensionnel en variables 0-1.” Rapport de recherche n∘1–2002, Université de Valenciennes et du Hainaut-Cambrésis, France.

  • Fréville, A. et G. Plateau. (1982). “Méthodes heuristiques performantes pour les programmes en variables 0-1.” Working paper, ANO-91, Université de Lille 1, France.

  • Fréville, A. and G. Plateau. (1986). “Heuristics and Reduction Methods for Multiple Constraints 0-1 Linear Programming Problems.” European Journal of Operational Research, 24, 206–215.

    Article  Google Scholar 

  • Fréville, A. and G. Plateau. (1990). “Hard 0-1 Multiknapsack Test Problems for Size Reduction Methods.” Investigacion Operativa 1, 251–270.

    Google Scholar 

  • Fréville, A. and G. Plateau. (1993a). “Sac-à-dos Multidimensionnel en Variables 0-1: Encadrement de la somme des Variables à l'optimum.” RAIRO Operations Research 27, 169–187.

    Google Scholar 

  • Fréville, A. and G. Plateau. (1993b). “An Exact Search for the Solution of the Surrogate Dual of the 0-1 Bidimensional Knapsack Problem.” European Journal of Operational Research 68, 413–421.

    Article  Google Scholar 

  • Fréville, A. and G. Plateau. (1994). “An Efficient Preprocessing Procedure for the Multidimensional Knapsack Problem.” Discrete Applied Mathematics 49, 189–212.

    Article  Google Scholar 

  • Fréville, A. and G. Plateau. (1996). “The 0-1 Bidimensional Knapsack Problem: Towards An Efficient High-Level Primitive Tool.” Journal of Heuristics 2, 147–167.

    Article  Google Scholar 

  • Frieze, A.M. and M.R. Clarke. (1984). “Approximate Algorithms for the m-Dimensional 0-1 Knapsack Problem: Worst-Case and Probabilistic Analyses.” European Journal of Operational Research 15, 100–109.

    Article  Google Scholar 

  • Gabrel, V., A. Knippel, and M. Minoux. (1999). “Exact Solutions of Multicommodity Network Optimization Problems with General Step Cost Functions.” Operations Research Letters 25, 15–23.

    Article  Google Scholar 

  • Gabrel, V. and M. Minoux (2002). “A Scheme for Exact Separation of Extended Cover Inequalities and Application to Multidimensional Knapsack Problem.” Operations Research Letters.

  • Gavish, B. and H. Pirkul. (1985). “Efficient Algorithms for Solving Multiconstraint Zero-One Knapsack Problems to Optimality.” Mathematical Programming 31, 78–105.

    Google Scholar 

  • Gens, G.V. and E.V. Levner. (1979). “Complexity and Approximation Algorithms for Combinatorial Problems: A Survey.” Central economic and Mathematical Institute, Academy of Sciences of the USSR, Moscow.

  • Geoffrion, A.M. (1974). “The Lagrangean Relaxation for Integer Programming.” Mathematical Programming Study 2, 82–114.

    Google Scholar 

  • van de Gerr, S., L. Stougie. (1991). “On Rates of Convergence and Asymptotic Normality in the Multiknapsack Problem.” Mathematical Programming 51, 349–358.

    Article  Google Scholar 

  • Gilmore P.C. and R.E. Gomory. (1966). “The Theory and Computation of Knapsack Functions.” Operations Research 14, 1045–1075.

    Google Scholar 

  • Glover, F. (1965). “A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem.” Operations Research 13, 879–919.

    Google Scholar 

  • Glover, F. (1968). “Surrogate Constraints.” Operations Research 16, 741–749.

    Google Scholar 

  • Glover, F. (1975). “Surrogate Constraints Duality in Mathematical Programming.” Operations Research 23, 434–451.

    Google Scholar 

  • Glover, F. (1977). “Heuristics for Integer Programming Using Surrogate Constraints.” Decision Sciences 8, 156–166.

    Google Scholar 

  • Glover, F. (1986). “Future Paths for Integer Programming and Links to Artificial Intelligence.” Computers and Operations Research 19, 533–549.

    Article  Google Scholar 

  • Glover, F. (1989). “Tabu Search, Part 1.” ORSA Journal on Computing 1, 190–206.

    Google Scholar 

  • Glover, F. (2001). “Tutorial on Surrogate Constraint Approaches for Optimization in Graphs.” Working paper, Hearin Center for Enterprise Science, University of Mississippi, USA.

  • Glover, F. (2003). “Exploiting Nested Inequalities and Surrogate Constraint.” Private communication.

  • Glover, F. and G.A. Kochenberger. (1996). “Critical Event Tabu Search for Multidimensional Knapsack Problems.” In I.H. Osman and J.P. Kelly (eds.), Meta-heuristics: Theory and Applications. Kluwer Academic Publishers, pp. 407–427.

  • Glover, F., Sherali H, and Y. Lee. (1997). “Generating Cuts from Surrogate Constraint Analysis for Zero-One and Multiple Choice Programming.” Computational Optimization and Applications 8, 154– 184.

    Article  Google Scholar 

  • Gotlieb, J. (1999). “On the Effectivity of Evolutionary Algorithms for Multidimensional Knapsack Problem.” Proceedings of Artificial Evolution (forth coming).

  • Green, C.J. (1967). “Two Algorithms for Solving Independent Multidimensional Knapsack Problems.” Management Sciences Report n ∘ 110, Carnegie institute of Technology, Graduate School of Industrial Administration, Pittsburgh, USA.

  • Greenberg, H. and W. Pierskalla. (1970). “Surrogate Mathematical Programs.” Operations Research 18, 924–939.

    Google Scholar 

  • Guignard, M. and S. Kim. (1987a). “Lagrangean Decomposition: A Model Yielding Stronger Lagrangean Bounds.” Mathematical Programming Study 39, 215–228.

    Google Scholar 

  • Guignard, M. and S. Kim. (1987b). “Lagrangean Decomposition for Integer Programming: Theory and Applications.” RAIRO 21, 307–323.

    Google Scholar 

  • Guignard, M. and K. Spielberg. (1969). “Search Techniques with Adaptive Features for Certain Integer and Mixed Integer Programming Problems.” Proceedings Information Processing 1968, Edinburgh, North Holland Publishing Company, pp. 238–244.

  • Guignard, M. and K. Spielberg. (1981), “Logical Reduction Methods in Zero-One Programming (Minimal Preferred Variables).” Operations Research 29, 49–74.

    Google Scholar 

  • Hammer, P., M. Padberg, and U. Peled. (1975). “Constraint Pairing in Integer Programming.” INFOR 13, 68–81.

    Google Scholar 

  • Hanafi, S. (1993). “Contribution à la Résolution de Problèmes Duaux de Grande Taille en Optimisation Combinatoire.” PhD thesis, University of Valenciennes, France.

  • Hanafi, S. and A. Fréville. (1998). “An Efficient Tabu Search Approach for the 0-1 Multidimensional Knapsack Problem.” European Journal of Operational Research 106, 659–675.

    Article  Google Scholar 

  • Hanafi, S. and A. Fréville (2001). “Extension of Reverse Elimination Method Through a Dynamic Management of the Tabu List.” RAIRO Operations Research 35, 251–267.

    Article  Google Scholar 

  • Haul, C. and S. Voss. (1998). “Using Surrogate Constraints in Genetic Algorithms for Solving Multidimensional Knapsack Problems.” in D.L. Woodruff (ed.), Advances in Computational and Stochastic Optimization, Logic Programming, and Heuristic Search, Kluwer Academic Publishers, pp. 235–251.

  • Hill, R. and C. Reilly (2000). “The Effects of Coefficient Correlation Structure in Two-Dimensional Knapsack Problems on Solution Procedure Performance.” Management Sciences 46, 302–317.

    Article  Google Scholar 

  • Hillier, F.S. (1969). “Efficient Heuristic Procedures for Integer Linear Programming with an Interior.” Operations Research 17, 600–637.

    Google Scholar 

  • Ibaraki, T. (1987). “Enumerative Approaches to Combinatorial Optimization—Part II.” Annals of Operations Research vol. 11.

  • Ibarra, O.H. and C.E. Kim. (1975). “Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems.” Journal of Association for Computing Machinery 22, 463–468.

    Google Scholar 

  • [97] IBM Corporation. (1990). Optimization subroutine library, Guide and Reference.

  • Ibaraki, T. and S. Isaka. (1983). “Double Relaxation Dynamic Programming Methods for the Multi-Dimensional Knapsack Problem.” Master Thesis, Department of Applied Mathematics and Physics, Kyoto University, Japan.

  • Johnson, E.J., G.L. Nemhauser, and M.W.P. Savelsbergh (2000). “Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition.” INFORMS Journal on Computing 12, 2–23.

    Article  Google Scholar 

  • Karwan, M.H. and R.L. Rardin. (1979). “Some Relationships Between Lagrangian and Surrogate Duality in Integer Programming.” Mathematical Programming 17, 320–334.

    Article  Google Scholar 

  • Karwan, M.H. and R.L. Rardin. (1980). “Searchability of the Composite and Multiple Surrogate Dual Functions.” Operations Research 28, 1251–1257.

    Google Scholar 

  • Karwan, M.H. and R.L. Rardin. (1984). “Surrogate Dual Multiplier Search Procedures in Integer Programming.” Operations Research 32, 52–69.

    Google Scholar 

  • Karwan, M.H. R.L. Rardin, and S. Sarin. (1987). “A New Surrogate Dual Multiplier Search Procedure.” Naval Research Logistics 34, 431–450.

    Google Scholar 

  • Khuri, S., T. Back, and J. Heitkotter. (1994). “The Zero/One Multiple Knapsack Problem and Genetic Algorithms.” Proceedings of the 1994 ACM Symposium on Applied Computing (SAC'99), ACM Press, pp. 188–193.

  • Kianfar, F. (1971). “Stronger Inequalities for 0-1 Integer Programming Using Knapsack Constraints.” Operations Research 19, 1374–1392.

    Google Scholar 

  • Kleinmuntz, D.N. and C.E. Kleinmuntz. (2001). “Multiobjective Capital Budgeting in Not-for-Profit Hospitals and Healthcare Systems.” Working paper, University of Illinois, Urbana-Champaign, USA, October 2001.

  • Kochenberger, G., G. McCarl, and F. Wyman. (1974). “A Heuristic for General Integer Programming.” Decision Sciences 5, 36–44.

    Google Scholar 

  • Korte, B. and R. Schnader. (1980). “On the Existence of Fast Approximate Schemes.” In O.L. Mangasarian, R.R. Meyer, S.M. Robinson (eds.), Nonlinear Programming Academic Press, pp. 415–437.

  • Korutcheva, E., M. Opper, and B. Lopez. (1994). “Statistical Mechanics of the Knapsack Problem.” Journal of Physics A: Mathematics and General 27, L645–L650.

    Article  Google Scholar 

  • Kwak, W., Y. Shi, H. Lee, and C.F. Lee. (1996). “Capital Budgeting with Multiple Criteria and Multiple Decision Makers.” Review of Quantitative Finance and Accounting 7, 97–112.

    Article  Google Scholar 

  • Laguna, M., J.P. Kelly, J.L. Gonzalez Velarde, and F. Glover. (1995). “Tabu Search for the Multilevel Generalized Assignment Problem.” European Journal of Operational Research 82, 176–189.

    Article  Google Scholar 

  • Lawler, E.L. (1979). “Fast Approximation Algorithms for Knapsack Problems.” Mathematics of Operations Research 4, 339–356.

    Google Scholar 

  • Lemke, C.E. and K. Spielberg. (1967). “Direct Search Algorithms for Zero-One and Mixed-Integer Programming.” Operations Research 15, 892–914.

    Google Scholar 

  • Lokketangen, A. and F. Glover. (1996). “Probabilistic Move Selection in Tabu Search for Zero-One Mixed Integer Programming Problems.” In I.H. Osman and J.P. Kelly (eds.), Meta-Heuristics: Theory & Applications Kluwer Academic Publishers, pp. 467–487.

  • Lokketangen, A. and F. Glover. (1998). “Solving Zero-One Mixed Integer Programming Problems Using Tabu Search.” Special Issue on Tabu Search, European journal of Operational Research 106, 624– 658.

    Article  Google Scholar 

  • Lokketangen, A. K. Jornsten, and S. Storoy. (1994), “Tabu Search Within a Pivot and Complement Framework.” International transactions of Operations Research 1, 305–316.

    Article  Google Scholar 

  • Lorie, J. and L.J. Savage. (1955). “Three Problems in Capital Rationing.” Journal of Business 28, 229– 239.

    Article  Google Scholar 

  • Magazine, M.J. and M.S. Chern. (1984). “A Note on Approximation Schemes for Multidimensional Knapsack Problems.” Mathematics on Operations Research 9, 244–247.

    Google Scholar 

  • Magazine, M.J. and O. Oguz. (1981). “A Fully Polynomial Approximation Algorithm for the 0-1 Knapsack Problem.” European Journal of Operational Research 8, 270–273.

    Article  Google Scholar 

  • Magazine, M.J. and O. Oguz. (1984). “A Heuristic Algorithm for the Multidimensional Zero-One Knapsack Problem”. European Journal of Operational Research 16, 319–326.

    Article  Google Scholar 

  • Mamer, J.W. and K.E. Schilling. (1990). “On the Growth of Random Knapsacks.” Discrete Applied Mathematics 28, 223–230.

    Article  Google Scholar 

  • Manne, A.S. and H.M. Markowitz. (1957). “On the Solution of Discrete Programming Problems.” Econometrica 25, 84–110.

    Google Scholar 

  • Marsten, R.E. and T.L. Morin. (1976). “MMPD, a Computer Code for Solving Multiconstraint Knapsack Problems in Integer Variables.” Working paper, Operations Research Center, MIT, Cambridge, USA.

  • Marsten, R.E. and T.L. Morin. (1977). “Optimal Solutions Found for Senju and Toyoda'0/1 Integer Programming Problems.” Management Sciences 23, 1364–1365.

    Google Scholar 

  • Marsten, R.E. and T.L. Morin. (1978). “A Hybrid Approach to Discrete Mathematical Programming.” Mathematical Programming 14, 21–40.

    Article  Google Scholar 

  • Martello, S. and P. Toth. (1988). “A New Algorithm for the 0-1 Knapsack Problem.” Management Sciences 34, 633–644.

    Google Scholar 

  • Martello, S. and P. Toth. (1990). “Knapsack Problems: Algorithms and Computer implementations.” In Series in Discrete Mathematics and Optimization, New York. Wiley InterSciences.

    Google Scholar 

  • Martello, S. and P. Toth. (1997). “Upper Bounds and Algorithms for Hard 0-1 Knapsack Problems.” Operations Research 45, 768–778.

    Google Scholar 

  • Martello, S., D. Pisinger, and P. Toth. (1999). “New Trends in Exact Algorithms for the 0-1 Knapsack Problem.” European Journal of Operational Research 123, 325–336.

    Article  Google Scholar 

  • Meanti, M., A.H.G. Rinnoy Kan, L. Stougie, and C. Vercellis. (1990). “A Probabilistic Analysis of the Multiknapsack Value Function.” Mathematical Programming 46, 237–247.

    Article  Google Scholar 

  • Meier, H., N. Christofides, and G. Salkin (2001). “Capital Budgeting Under Uncertainty—an Integrated Approach Using Contingent Claims Analysis and Integer Programming.” Operations Research 49, 196–206.

    Article  Google Scholar 

  • Nediak, M. and J. Eckstein (2001). “Pivot, Cut and Dive: A Heuristic for 0-1 Mixed Integer Programming.” Rutcor research report, Rutgers University, USA, October 2001.

  • Nemhauser, G.L., M.W.P. Savelsbergh, and G.C. Sigismondi. (1993). “MINTO, a Mixed INTeger Optimizer.” Operations Research Letters 15, 47–58.

    Article  Google Scholar 

  • Nemhauser, G.L. and L.A. Wolsey. (1988). “Integer and Combinatorial Optimization.” John Wiley and Sons.

  • Niar, S. and A. Fréville. (1997). “A Parallel Tabu Search Algorithm for the 0-1 Multidimensional Knapsack Problem.” International Parallel Processing Symposium'97, IEEE Computer Society Press, pp. 512–516, Suisse.

  • Ohlssen, M., C. Peterson, and B. Soderberg. (1993). “Neural Networks for Optimization Problems with Inequality Constraints: The Knapsack Problem.” Neural Computation 5, 331–339.

    Google Scholar 

  • Osorio, M.A., F. Glover, and P. Hammer (2002). “Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions.” Annals of Operations Research 117, 71–93.

    Article  Google Scholar 

  • Pirkul, H. (1987). “A Heuristic Solution Procedure for the Multiconstraint Zero-One Knapsack Problem.” Naval Research Logistics 34, 161–172.

    Google Scholar 

  • Pisinger, D. (1995). “An Expanding-Core Algorithm for the Exact 0-1 Knapsack Problem.” European Journal of Operational Research 87, 175–187.

    Article  Google Scholar 

  • Pisinger, D. (2000). “A Minimal Algorithm for the Bounded Knapsack Problem.” ORSA Journal on Computing 12, 75–84.

    Google Scholar 

  • Plateau, A., D. Tachat, and P. Tolla (2001). “A Hybrid Search Combining Interior Point Methods and Metaheuristics for 0-1 Programming.” in Investigacion Operativa forth coming.

  • Plateau, G. (1976). “Réduction de la Taille Des Problèmes Linéaires en Variables 0-1.” Working paper, ANO-71, Université de Lille 1, France.

  • Plateau, G. and M. Elkihel. (1985). “A Hybrid Method for the 0-1 Knapsack Problem.” Methods of Operations Research 49, 277–293.

    Google Scholar 

  • Plateau, G. and C. Roucairol. (1989). “A Supercomputer Algorithm for the 0-1 Multiknapsack Problem.” In R. Sharda, B. Golden, E. Wasil, O. Balei and W. Stewart, (eds.), Impacts of Recent Computer Advances on Operations Research, Operations Research Series pp. 144–157.

  • Raidl G.R. (1998). “An Improved Genetic Algorithm for the Multiconstraint Knapsack Problem.” in Proceedings of the 5 th IEEE International Conference on Evolutionary Computation pp. 207–211.

  • Reilly, C.H. (1991). “Optimization Test Problems with Uniformly Distributed Coefficients.” Proceedings of the 1991 Winter Simulation Conference, (eds.), B.L. Nelson, W.D. Kelton and G.M. Clark, Institute of Electrical and Electronics Engineers, Phoenix, USA, 866–874.

  • Rinnoy, Kan A.H.G., L. Stougie, and C. Vercellis. (1993). “A Class of Generalized Greedy Algorithms for the Multi-Knapsack Problem.” Discrete applied Mathematics 42, 279–290.

    Article  Google Scholar 

  • Savelsbergh, M.W.P. (1994). “Preprocessing and Probing Techniques for Mixed Integer Programming Problems.” ORSA Journal of Computing 6, 445–454.

    Google Scholar 

  • Schilling, K.E. (1990). “The Growth of m-Constraint Random Knapsacks.” European Journal of Operational Research 46, 109–112.

    Article  Google Scholar 

  • Schilling, K.E. (1994). “Random Knapsacks with Many Constraints.” Discrete Applied Mathematics 48, 163–174.

    Article  Google Scholar 

  • Senju, S. and Y. Toyada. (1968). “An Approach to Linear Programming Problems with 0-1 Variables.” Management Sciences 15, 196–207.

    Article  Google Scholar 

  • Shani, S. (1975). “Approximate Algorithms for the 0-1 Knapsack Problem.” Journal of Association for Computing Machinery 22, 115–124.

    Google Scholar 

  • Shih, W. (1979). “A Branch and Bound Method for the Multiconstraint Zero-One Knapsack Problem.” Journal of Operations Research Society 30, 369–378.

    Google Scholar 

  • Soyster, A.L., B. Lev and W. Slivka. (1978). “Zero-One Programming with Many Variables and Few Constraints.” European Journal of Operational Research 2, 195–201.

    Article  Google Scholar 

  • Szkatula, K. (1994). “The Growth of Multi-Constraint Random Knapsacks with Various Right-Hand Sides of the Constraints.” European Journal of Operational Research 73, 199–204.

    Article  Google Scholar 

  • Szkatula, K. (1997). “The Growth of Multi-Constraint Random Knapsacks with Large Right-Hand Sides of the Constraints.” Operations Research Letters 21, 25–30.

    Article  Google Scholar 

  • Szkatula, K. (2000). “How Fast the Random Knapsacks with Many Constraints Grow.” Working paper presented at ECCO XIII conference, Capri, Italy.

  • Thesen, A. (1975). “A Recursive Branch and Bound Algorithm for the Multidimensional Knapsack Problem.” Naval Research Quaterly 22, 341–353.

    Google Scholar 

  • Thiel, J. and S. Voss. (1994). “Some Experiences on Solving Multiconstraint Zero-One Knapsack Problems with Genetic Algorithms.” INFOR 32, 226–242.

    Google Scholar 

  • Toulouse, M., T.G. Crainic, and M. Gendreau. (1996). “Communication Issues in Designing Cooperative Multithread Parallel Searches.” In I.H. Osman and J.P. Kelly (eds.), Meta-Heuristics : Theory and Applications Kluwer Academic Publishers, pp. 503–522.

  • [177] Toyoda, Y. (1975). “A Simplified Algorithm for Obtaining Approximate Solutions to Zero-One Programming Problems.” Management Sciences 21, 1417–1427.

    Google Scholar 

  • Vasquez, M. and J.K. Hao (2001a). “A Logic-Constrained Knapsack Formulation and a Tabu Algorithm for the Daily Photograph Scheduling of an Earth Observation Satellite.” Journal of Computational Optimization and Applications forth coming.

  • Vasquez, M. and J.K. Hao (2001b). “Une Approche Hybride Pour le Sac à dos Multidimensionnel en Variables 0-1.” in RAIRO Operations Research forthcoming.

  • Voss, S. (1993). “Tabu Search: Applications and Prospects.” In: D.Z. Du and P.M. Pardalos, (eds.), Network Optimization Problems pp. 333–353.

  • Weingartner, H.M. and D.N. Ness. (1967). “Method for the Solution for the Multidimensional 0-1 Knapsack Problem.” Operations Research 15, 83–103.

    Google Scholar 

  • Wolsey, L.A. (1980). “Heuristic Analysis, Linear Programming and Branch and Bound.” Mathematical Programming Study 13, 121–134.

    Google Scholar 

  • Yang, J. (1994). “A Computational Study on 0-1 Knapsack Problems Generated Under Explicit Correlation Induction.” MS Thesis, Department of Industrial and Systems Engineering, The Ohio State University, Columbus, USA.

Download references

Author information

Authors and Affiliations

  1. Laboratoire d'Automatique, de Mécanique et d'Informatique Industrielles et Humaines, UMR CNRS 8530, Groupe Recherche Opérationnelle et Informatique, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313, Valenciennes Cedex, France

    Arnaud Fréville & SaÏd Hanafi

Authors
  1. Arnaud Fréville
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. SaÏd Hanafi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arnaud Fréville.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fréville, A., Hanafi, S. The Multidimensional 0-1 Knapsack Problem—Bounds and Computational Aspects. Ann Oper Res 139, 195–227 (2005). https://doi.org/10.1007/s10479-005-3448-8

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-005-3448-8

Keywords

Search

Navigation