Stability of utility functions and apportionment rules in location models
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Stability of utility functions and apportionment rules in location models H. A. Eiselt1 · Vladimir Marianov2 Received: 11 February 2020 / Accepted: 12 May 2020 © Sociedad de Estadística e Investigación Operativa 2020
Abstract This paper investigates how demand is apportioned to facilities by customers, given that they apply one of a number of utility functions according to which they satisfy their demand at the facilities. After delineating the basic decision-making process, a number of reasonable assumptions are formulated regarding the behavior of the utility functions after a scaling of their different parameters. The individual apportionment rules are examined so as to whether or not they satisfy these assumptions. The results are of importance for decision-makers that must use these utility functions when facing locational decisions. Keyowords Competitive location · Market share · Customers’ preferences · Apportionment rules analysis Mathematics Subject Classification 90B80 · 91B42
1 Setting the stage: customer‑facility interactions in location models Location models have arguably been discussed and solved since Fermat’s posting of unsolved problems in 1640 and Torricelli’s reply in 1646 (see, e.g., Eckhardt 2008). There are hundreds, if not thousands of location models that have been discussed in the literature. As a case in point, Trevor Hale’s (2015) listing contains no less than 3400 references, even though most entries are more than 20 years old. There are multitudes of different models, applications, and theorems, but all of them have in common that they include three main components: * Vladimir Marianov [email protected] H. A. Eiselt [email protected] 1
Faculty of Business Administration, University of New Brunswick, Fredericton, NB, Canada
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Department of Electrical Engineering, Instituto Sistemas Complejos de Ingeniería (ISCI), Pontificia Universidad Católica de Chile, Santiago, Chile
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• a space, in which all interactions between the players take place, • customers at (almost always) fixed locations with some demand for a good or a
service, and
• facilities, which are to be located by one or more firms.
One may image the huge number of different models with different configurations of the above main elements, their interactions, and myriads of other features. To introduce some order, various authors have suggested taxonomies. Prominent examples are Nickel and Puerto (2005) for location problems in general and Eiselt et al. (1993) for competitive location models. Interactions between the agents in location models (viz., customers and facilities) may take many different forms: in physical spaces, facilities may ship goods to customers, customers may travel to the facilities to obtain a good or a service, or similar. There are, however, also nonphysical spaces. The most popular examples include brand positioning and political models. In these models, a space is defined that includes all quantitative features of a class of products or se
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