Technology Category
Constraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationSearch and Optimization: Mixed Discrete/Continuous SearchPlanning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
Investigating mixed-integer programming approaches for the $p$-$α$-closest-center problem
🤖 AI Summary
This study addresses the $p$-$\alpha$-closest center problem ($p\alpha$CCP), which seeks to select $p$ facilities such that the maximum, over all clients, of the sum of distances to their $\alpha$ nearest facilities is minimized. The authors propose four mixed-integer programming (MIP) formulations—the first of their kind for this problem—and conduct a polyhedral analysis to compare the strength of their linear relaxations. They further enhance these models through iteratively lifted inequalities. Building upon this foundation, they develop the first exact branch-and-cut algorithm for $p\alpha$CCP, integrating variable fixing and heuristic strategies. Computational experiments demonstrate the algorithm’s superiority over existing variable neighborhood search methods, with optimal solutions proven for 17 out of 40 benchmark instances.
Technology Category
Application Category
📝 Abstract
In this work, we introduce and study the $p$-$α$-closest-center problem ($pα$CCP), which generalizes the $p$-second-center problem, a recently emerged variant of the classical $p$-center problem. In the $pα$CCP, we are given sets of customers and potential facility locations, distances between each customer and potential facility location as well as two integers $p$ and $α$. The goal is to open facilities at $p$ of the potential facility locations, such that the maximum $α$-distance between each customer and the open facilities is minimized. The $α$-distance of a customer is defined as the sum of distances from the customer to its $α$ closest open facilities. If $α$ is one, the $pα$CCP is the $p$-center problem, and for $α$ being two, the $p$-second-center problem is obtained, for which the only existing algorithm in literature is a variable neighborhood search (VNS).
We present four mixed-integer programming (MIP) formulations for the $pα$CCP, strengthen them by adding valid and optimality-preserving inequalities and conduct a polyhedral study to prove relationships between their linear programming relaxations. Moreover, we present iterative procedures for lifting some valid inequalities to improve initial lower bounds on the optimal objective function value of the $pα$CCP and characterize the best lower bounds obtainable by this iterative lifting approach.
Based on our theoretical findings, we develop a branch-and-cut algorithm (B&C) to solve the $pα$CCP exactly. We improve its performance by a starting and a primal heuristic, variable fixings and separating inequalities. In our computational study, we investigate the effect of the various ingredients of our B&C on benchmark instances from related literature. Our B&C is able to prove optimality for 17 of the 40 instances from the work on the VNS heuristic.
Problem
Research questions and friction points this paper is trying to address.
p-center problem
mixed-integer programming
facility location
α-distance
combinatorial optimization
Innovation
Methods, ideas, or system contributions that make the work stand out.
mixed-integer programming
p-α-closest-center problem
branch-and-cut algorithm
polyhedral study
valid inequalities
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