Optimal Worst-Case Matching

Definition

Let P be a set of service providers and O be a set of customers. Each service-provider p (customer o) has a service capacity (demand), denoted by p. w (o. w). The Euclidean distance between o and p is denoted by d(o, p). The SPatialMatching forMinimizingMaximum matching distance (SPM-MM) problem (Long et al. 2013) is to generate an assignment A denoting a set containing the elements in the form of triplets (o, p, w), where (o, p, w) is called a match between o and p and denotes that p provides the service with the amount of w to o such that the following three conditions hold.

• (1) Capacity Constraint: no service provider provides its service of the amount greater than its capacity, i.e., $\forall p \in P$, ∑_{(o, p, w) ∈ A} w ≤ p. w;

• (2) Demand Constraint: each customer’s service demand is satisfied, i.e., $\forall o \in O$, ∑_{(o, p, w) ∈ A}w = o. w; and

• (3) Optimality Constraint: the maximum matching distance (mmd) of A is minimized, i.e., max{d(o, p) | (o, p, w) ∈ A}, is...