Location-Allocation Modeling

Though often used to refer to more general location-theoretic constructs, location-allocation modeling is a specific class of spatial optimization model where two simultaneous decisions are being made about where a facility should be sited (location) and what entities/areas should be served by the facility (allocation). Geographers have made important and sustained contributions to this area of research, developing and extending mathematical models, devising effective solution techniques, and applying models to public and private sector planning problems. Prominent geographers working in the area include Charles ReVelle, Gerard Rushton, Michael Goodchild, and Richard Church, among others.

A location-allocation model locates a multiple number of facilities and allocates the demand served by these facilities so that access and/or system service is as efficient as possible. This definition recognizes the significance and necessity of optimizing a system on the whole in a coordinated fashion, in contrast to one-at-a-time addition of a facility when multiple facilities are sited.

Many different types of facilities are possible in a location-allocation model. In retail, a facility could be an outlet, store, warehouse, restaurant, and so on. Facilities associated with emergency services could include not only fire and police stations but also ambulances, warning sirens, disaster relief centers, and so on. Potential facilities could be schools, libraries, or even public pools and salt pile storage sites for road maintenance. Depending on the type of facility, examples of demand to be served could be stores receiving merchandise from a warehouse, neighborhoods being serviced by a school, or homes/businesses being protected by a fire station. Given the facility and demand types, the location-allocation problem involves selecting facility sites (location) and prescribing which demand is served by what facility (allocation). Of interest is accomplishing the location and allocation in such a manner that system service is as good and/or efficient as possible. System service then is characterized in terms of economic efficiency.

A location-allocation model is a mathematical representation of a planning problem consisting of decision variables associated with location and allocation choices, the objective(s) to be optimized, and the constraining conditions that must be satisfied. The model is specified in algebraic terms as a series of linear and/or nonlinear functions. Some prominent location-allocation models include the simple plant location problem, the p-median problem, the capacitated plant location problem, and the transportation p-median problem. Of course, there are many other location-allocation models, some extensions of these basic problems.

There are two primary variants of location-allocation models, distinguishable by their spatial representation of potential facility locations. One variant allows facilities to be located anywhere in continuous space. Facilities are therefore considered to be feasible at any location in space. Travel through space is often assumed to occur in a readily defined manner, such as Euclidean or rectilinear distance. In contrast, the other primary variant assumes that facilities are only allowed at discrete locations. That is, facilities are limited to a finite set of potential sites. As a result, travel through space is assumed to take place through a transportation network consisting of nodes and arcs. Other distinctions of location-allocation models include capacitated versus uncapacitated facilities, stochastic versus static demand, and complete versus partial assignment of demand to a facility.

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