Constructs for Multilevel Closest Assignment in Location Modeling

Abstract

In the classic p-median problem, it is assumed that each point of demand will be served by his or her closest located facility. The p-median problem can be thought of as a ‘‘single-level’’ allocation and location problem, as all demand at a specific location is assigned as a whole unit to the closest facility. In some service protocols, demand assignment has been defined as ‘‘multilevel’’ where each point of demand may be served a certain percentage of the time by the closest facility, a certain percentage of the time by the second closest facility, and so on. This article deals with the case in which there is a need for ‘‘explicit’’ closest assignment (ECA) constraints. The authors review past location modeling work that involves single-level ECA constraints as well as specific constraint constructs that have been proposed to ensure single-level closest assignment. They then show how each of the earlier proposed ECA constructs can be generalized for the ‘‘multilevel’’ case. Finally, the authors provide computational experience using these generalized ECA constructs for a novel multilevel facility interdiction problem introduced in this article. Altogether, this article proposes both a new set of constraint structures that can be used in location models involving multilevel assignment as well as a new facility interdiction model that can be used to optimize worst case levels of facility disruption.

When centers can fail: A close second opportunity

“…In this formulation this is done by means of Closest Assignment Constraints (CAC) (4). Recent studies of CAC for standard discrete location formulations can be consulted in [16,22]. Actually, constraints (4) can be replaced by the stronger constraints…”

“…Note that variables y jk with j ak only appear in one term of constraints (32) that is required to cope correctly with situations where different centers can be set at the same distance from a given site i, and in the linking constraints (21) and (22) required to force them to take values consistent with those of variables y j . Thus, if there are no ties in the distances from any site i to the others, this formulation can be sensibly reduced.…”

Location Science

“…Losada et al (2012b) further extended this probabilistic r-IMP by assuming that the probability of impairing a facility depends on the intensity of the disruption or on the amount of offensive resources used in the attack. In a further extension, Lei and Church (2011) address the issue of interdiction when not all demands are served by their closest facility after a disruption.…”

Distribution of the Ratio of Distances to the First and Second Nearest Facilities

“…They made an ordered assignment of customers to facilities. Lei and Church [8] presented generalized closest assignment constraints in terms of multiple levels of closeness. Miyagawa [13] found the optimal location that minimizes the average distance to the nearest open facility when some of the existing facilities are closed.…”