Stochastic Location-Assignment on the Unit Interval
Abstract
This dissertation is composed of two self-contained essays on stochastic location-assignment on the unit interval. In a stochastic location-assignment model a given number of servers are positioned on the unit interval to serve demands with random locations. The total number of demands may not be known apriori, but will not exceed the number of servers. Two kinds of demand arrivals are considered: (i) sequential; (ii) simultaneous. Chapter 1 addresses demands that arrive sequentially. Upon arrival of each demand a server is assigned. The assignment cost for a server-demand pair is the distance between them. Once a server is assigned it is unavailable to serve any subsequent demand. The objective is to determine the server-locations which minimize the expected value of the total assignment cost. 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 objective function and the uniqueness of optimal server-locations are shown. Finally, some insights are derived, based on numerical results, on the spatial distribution of the optimal server-locations for specific instances. Chapter 2 addresses demands that arrive simultaneously. A server is assigned to each demand with no server assigned to more than one demand. The assignment cost for a server-demand pair is the distance between them. The objective is to locate the servers so as to minimize the expected value of the total assignment cost. A necessary condition for optimality of servers-locations is derived. Specific subclasses of the model are considered; for which the number of demands is known apriori. For number of demands equal to the number of servers, the optimal assignment policy has an explicit form, and the optimal server-locations are the medians of the order statistic of demands. For two demands whose locations are independent and uniformly distributed over the interval, the convexity of the objective function and the uniqueness of optimal server-locations are shown. Finally, some insights are derived, based on numerical results, on the spatial distribution of the optimal server-locations for specific instances.
Degree
Ph.D.
Advisors
Ward, Purdue University.
Subject Area
Management|Operations research|Urban planning
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