Abstract
In the assignment problem units of supply are assigned on a one-to-one basis to units of demand so as to minimize the sum of the cost associated with each supply-to-demand matched pair. Defined on a network, the supplies and demands are located at vertices and the cost of a supply-to-demand matched pair is the distance between them. This paper considers a two-stage stochastic program for locating the units of supply based upon only a probabilistic characterization of demand. The objective of the first-stage location problem is to minimize the expected cost of the second-stage assignment problem. Principal results include showing that the problem is NP-hard on a general network, has a simple solution procedure on a line network, and is solvable by a low order polynomial greedy procedure on a tree network. Potential applications are discussed.
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References
Ahuja, R.K., T.L. Magnanti, J.B. Orlin, and M.R. Reddy. (1995). Chapter 1: Applications of Network Optimization, Ball, M.O, T.L. Magnanti, C.L., Monma, G.L. Nemhauser, Network Models, Amsterdam: North-Holland, 1–75.
Batta, R., R.C. Larson, and A.R. Odoni. (1988). “A Single Server Priority Queuing-Location Model.” Networks 18, 87–103.
Berman, O., R.C. Larson, and S.S. Chiu. (1985a). “Optimal Server Location on a Network Operating as an M/G/1 Queue.” Operations Research 33(4), 746–771.
Berman, O., R.C. Larson, and S.S. Chiu. (1985b). “Locating a Mobile Server queuing Facility on a Tree Network.” Management Science 31(6), 764–772.
Birge, J.R. and F. Louveaux. (1997). Introduction to Stochastic Programming. Springer.
Brandeau, M.L. and S.S. Chiu. (1990). “A Unified Family of Single Server Queuing Location Models.” Operations Research 38(6), 1034–1044.
Brandeau, M.L. and S.S. Chiu. (1989). “An Overview of Representative Problems in Location Research.” Management Science 35(6), 645–674.
Erlenkotter, D. (1978). “A Dual-Based Procedure for Uncapacitated Facility Location Problems.” Operations Research 26(6), 992–1009.
Franca, P.M. and H.P.L. Luna. (1982). “Solving Stochastic Transportation Location Problems by Generalized Benders Decomposition.” Transportation Science 16(2), 113–126.
Garey, M.R. and D.S. Johnson. (1979). Computer and Intractability. W. H. Freeman.
Goldman, A.J. (1971). “Optimal Center Location in Simple Networks.” Transportation Science 5, 212–221.
Hakimi, S.L. (1964). “Optimum Location of Switching Centers and the Absolute Centers and Medians of a Graph.” Operations Research 12, 450–459.
Kariv, O. and S.L. Hakimi. (1979). “An Algorithmic Approach to Network Location Problems, Part II.” SIAM J. Applied Math. 37, 539–560.
Laporte, G. and F. Louveaux. (1993). “The integer L-shaped Method for Stochastic Integer Programs with Complete Recourse.” Operations Research Letters 13(3), 133–142.
Laporte, G., F. Louveaux, and L. Van Hamme. (1994). “Exact Solution to a Location Problem with Stochastic Demands.” Transportation Science 28, 95–103.
Louveaux, F.V. and D. Peeters. (1992). “A Dual-based Procedure for Stochastic Facility Location.” Operations Research 40(3), 564–573.
Louveaux, F.V. and J.F. Thisse. (1985). “Production and Location on a Network Under Demand Uncertainty.” Operations Research Letter 4(4), 145–149.
Papadimitriou, C.H. and K. Steiglitz. (1978). Combinatorial Optimization. Prentice Hall.
Prasad, S.Y. and R. Batta. (1993). “Determining Efficient Facility Locations on a Tree Network Operating as FIFO M/G/1 Queue.” Networks 23, 597–603.
Wallace, S.W. (1986). “Solving Stochastic Programs with Network Recourse.” Networks 16, 295–317.
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Zeng, T., Ward, J.E. The Stochastic Location-Assignment Problem on a Tree. Ann Oper Res 136, 81–97 (2005). https://doi.org/10.1007/s10479-005-2040-6
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DOI: https://doi.org/10.1007/s10479-005-2040-6