Abstract
A given number of servers are pre-positioned on an interval to serve demands that will arrive at random locations on the interval. The total number of demands to be served may not be known apriori, but will not exceed the number of servers. The demands arrive sequentially, and upon arrival, each is assigned to one of the servers, with that server unavailable for future demands. The remaining unassigned servers are not repositioned. The cost associated with each assigned server-demand pair is the distance between them. Both server-location and server-to-demand assignment decisions are made to minimize the expected sum of the costs for all assigned pairs. A necessary condition for optimality of servers-locations is derived, and the set of optimal servers-locations is characterized. For demands that are independent and uniformly distributed over the interval, the convexity of the location problem’s objective function and the uniqueness of optimal server-locations are shown. Optimal server locations are found for problems with up to 10 servers and uniform demands, and some insights are derived based on these instances.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson LR, Fontenot RA (1992) Optimal positioning of service units along a coordinate line. Transp Sci 26:346–351
Bartle RG (1976) The elements of real analysis, 2nd edn. Wiley, New York
Carrizosa E, Munoz-Marquez M, Puerto J (1998) A note on the optimal positioning of service units. Oper Res 46:155–156
Carrizosa E, Munoz-Marquez M, Puerto J (2002) Optimal positioning of a mobile service unit on a line. Ann Oper Res 111:75–88
Cavalier TM, Sherali HD (1985) Sequential location allocation problems on chains and trees with probabilistic link demands. Math Program 32:249–277
Cooper L (1963) Location-allocation problems. Oper Res 11:331–343
Derman C, Lieberman GJ, Ross SM (1972) A sequential stochastic assignment problem. Manage Sci 18:349–355
Huang Y, Fan Y, Johnson N (2009) Multistage system planning for hydrogen production and distribution. Published Online by Networks and Spatial Economics December 4, 2009
Leonhardt D (13th October 2005) Shipper’s Agility in Meeting Challenges Points the Way for U.S. Economy. The San Diego Union-Tribune
Levine A (1986) A patrol problem. Math Mag 59:159–166
Love RF (1976) One-dimensional facility location-allocation using dynamic programming. Manage Sci 22:614–617
Milgrom P, Roberts J (1990) Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58:1255–1277
Puerto J, Rodriguez-Chia AM (2003) Robust positioning of service units. Stoch Models 19:125–147
Rosenhead J, Powell G (1973) The ice-cream man problem. Transp Res 9:117–121
Sakaguchi M (1984) A sequential stochastic assignment problem with an unknown number of jobs. Math Jpn 29:141–152
Scott AJ (1970) Location-allocation systems: a review. Geogr Anal 2:95–119
Seber GAF (2008) A matrix handbook for statisticians (Wiley Series in Probability and Statistics). Wiley-Interscience, Hoboken
Sherali HD (1991) Capacitated, balanced, sequential location-allocation problems on chains and trees. Math Program 49:381–396
Smith DK (1997) Police patrol policies on motorways with unequal patrol lengths. J Oper Res Soc 48:996–1000
Topkis DM (1978) Minimizing a submodular function on a lattice. Oper Res 26:305–321
Wesolowsky GO (1977) Probabilistic weights in the one-dimensional facility location problem. Manage Sci 24:224–229
Zeng T, Ward JE (2005) The stochastic location-assignment problem on a tree. Ann Oper Res 136:81
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Viswanath, K., Ward, J. Stochastic Location-assignment on an Interval with Sequential Arrivals. Netw Spat Econ 10, 389–410 (2010). https://doi.org/10.1007/s11067-010-9137-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11067-010-9137-4