Abstract
The assignment problem is a well-known operations research model. Its various extensions have been applied to the design of distributed computer systems, job assignment in telecommunication networks, and solving diverse location, truck routing and job shop scheduling problems.
This paper focuses on a dynamic generalization of the assignment problem where each task consists of a number of units to be performed by an agent or by a limited number of agents at a time. Demands for the task units are stochastic. Costs are incurred when an agent performs a task or a group of tasks and when there is a surplus or shortage of the task units with respect to their demands. We prove that this stochastic, continuous-time generalized assignment problem is strongly NP-hard, and reduce it to a number of classical, deterministic assignment problems stated at discrete time points. On this basis, a pseudo-polynomial time combinatorial algorithm is derived to approximate the solution, which converges to the global optimum as the distance between the consecutive time points decreases. Lower bound and complexity estimates for solving the problem and its special cases are found.
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References
R.K. Ahuja T.L. Magnati J.B. Orlin (1993) Network Flows: Theory, Algorithms, and Applications Prentice-Hall Englewood Cliffs, NJ Occurrence Handle1201.90001
V.R. Algazi D.J. Sakrison (1969) ArticleTitleOn the Optimality of the Karhunen–Loève Expansion IEEE Transactions on Information Theory 15 319–321 Occurrence Handle10.1109/TIT.1969.1054286 Occurrence Handle258158 Occurrence Handle0175.46602
V. Balachandran (1976) ArticleTitleAn integer generalized transportation model for optimal job assignment in computer networks Operations Research 24 742–759 Occurrence Handle10.1287/opre.24.4.742 Occurrence Handle439170 Occurrence Handle0356.90028
E. Balas E. Zemel (1980) ArticleTitleAn algorithm for large zero–one Knapsack problems Operations Research 28 1130–1154 Occurrence Handle10.1287/opre.28.5.1130 Occurrence Handle589676 Occurrence Handle0449.90064
A.Y. Dubovitsky A.A. Milyutin (1981) Theory of the Maximum Principle V.L. Levin (Eds) Methods of Theory of Extremum Problems in Economy Nauka, Moscow 6–47
R.F. Hartl S.P. Sethi R.G. Vickson (1995) ArticleTitleA survey of the maximum principles for optimal control problems with state constraints SIAM Review 37 IssueID2 181–218 Occurrence Handle10.1137/1037043 Occurrence Handle1343211 Occurrence Handle0832.49013
Hillier F.S. and Lieberman, G.J. (1995), Introduction to Operations Research, McGraw-Hill.
M.R. Garey D.S. Johnson (1991) Computers and Intractability: A guide to the Theory of NP-Completeness W.H. Freeman and Company New York
B. Gavish H. Pirkul (1986) ArticleTitleComputer and database location in distributed computer systems IEEE Transactions on Computers 35 IssueID7 583–590 Occurrence Handle10.1109/TC.1986.1676799
B. Gavish H. Pirkul (1991) ArticleTitleAlgorithms for the multi-resource generalized assignment problem Management Science 37 IssueID6 695–713 Occurrence Handle10.1287/mnsc.37.6.695 Occurrence Handle0753.90053
T.D. Klastorin (1979) ArticleTitleAn effective subgradient algorithm for the generalized assignment problem Computers and Operations Research 6 155–164 Occurrence Handle10.1016/0305-0548(79)90028-5
E. Khmelnitsky K. Kogan O. Maimon (1995) ArticleTitleA maximum principle based combined method for scheduling in a flexible manufacturing system Discrete Events Dynamic Systems 5 343–355 Occurrence Handle10.1007/BF01439152 Occurrence Handle0837.90061
K. Kogan A. Shtub V. Levit (1997) ArticleTitleDGAP–The dynamic generalized assignment problem Annals of Operations Research 69 227–239 Occurrence Handle10.1023/A:1018933012422 Occurrence Handle0880.90076
S. Martello P. Toth (1977) ArticleTitleAn upper bound for the zero–one Knapsack problem and branch and bound Algorithm European Journal of Operational Research 1 169–175 Occurrence Handle10.1016/0377-2217(77)90024-8 Occurrence Handle452677 Occurrence Handle0374.90050
J.B. Mazzola A.W. Neebe (1986) ArticleTitleResource-constrained assignment scheduling Operations Research 34 IssueID4 560–572 Occurrence Handle10.1287/opre.34.4.560 Occurrence Handle874295 Occurrence Handle0609.90087
Murphy, R.A. (1986), A Private Fleet Model with Multi-Stop Backhaul, Working Paper, 103, Optimal Decision Systems, Green Bay, WI.
R.M. Nauss (1976) ArticleTitleAn efficient algorithm for the 0–1 Knapsack problem Management Science 23 27–31 Occurrence Handle10.1287/mnsc.23.1.27 Occurrence Handle0337.90042
G.T. Ross R.M. Soland (1975) ArticleTitleA branch and bound algorithm for the generalized assignment problem Math. Programming 8 91–103 Occurrence Handle10.1007/BF01580430 Occurrence Handle368757 Occurrence Handle0308.90028
G.T. Ross R.M. Soland (1977) ArticleTitleModeling facility location problems as generalized assignment problems Management Science 24 345–357 Occurrence Handle10.1287/mnsc.24.3.345 Occurrence Handle0378.90066
Sousa, J.B. and Pereira, F.L. (1992), A hierarchical framework for the optimal flow control in manufacturing systems. In: Proceedings of the Third International Conference on Computer Integrated Manufacturing , Troy, pp. 278–286.
Van Trees, H.L. (1968), Detection, Estimation and Modulation Theory, Part I, John Wiley Sons.
M. Yagiura T. Yamaguchi T. Ibaraki (1999) A variable depth search algorithm for the generalized assignment problem S. Voss S. Martello I.H. Osman C. Roucairol (Eds) Meta-heuristics: Advances and Trends in Local Search Paradigms for Optimization Springer US Boston 459–471 Occurrence Handle10.1007/978-1-4615-5775-3_31
Youla, D.C. (1957), The solution of a homogeneous Wiener–Hopf integral equation occurring in the expansion of second-order stationary random functions, IEEE Transactions on Information Theory, 187–193.
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Kogan, K., Khmelnitsky, E. & Ibaraki, T. Dynamic Generalized Assignment Problems with Stochastic Demands and Multiple Agent--Task Relationships. J Glob Optim 31, 17–43 (2005). https://doi.org/10.1007/s10898-004-4273-3
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DOI: https://doi.org/10.1007/s10898-004-4273-3