[Paper Review] Two-sided Facility Location

Reza Alijani, Siddhartha Banerjee+3 more|arXiv (Cornell University)|Nov 30, 2017

Transportation and Mobility Innovations13 references3 citations

TL;DR

This paper introduces Two-sided Facility Location, a novel optimization framework for e-commerce marketplaces that jointly determines pricing, facility placement, and agent routing to maximize social surplus under quality and availability constraints. It presents an LP rounding algorithm achieving a (1+ε) approximation to surplus for any ε>0, while relaxing match distance by a constant factor, with extensions to profit and throughput objectives.

ABSTRACT

Recent years have witnessed the rise of many successful e-commerce marketplace platforms like the Amazon marketplace, AirBnB, Uber/Lyft, and Upwork, where a central platform mediates economic transactions between buyers and sellers. Motivated by these platforms, we formulate a set of facility location problems that we term Two-sided Facility location. In our model, agents arrive at nodes in an underlying metric space, where the metric distance between any buyer and seller captures the quality of the corresponding match. The platform posts prices and wages at the nodes, and opens a set of facilities to route the agents to. The agents at any facility are assumed to be matched. The platform ensures high match quality by imposing a distance constraint between a node and the facilities it is routed to. It ensures high service availability by ensuring flow to the facility is at least a pre-specified lower bound. Subject to these constraints, the goal of the platform is to maximize the social surplus (or gains from trade) subject to weak budget balance, i.e., profit being non-negative. We present an approximation algorithm for this problem that yields a $(1 + ε)$ approximation to surplus for any constant $ε> 0$, while relaxing the match quality (i.e., maximum distance of any match) by a constant factor. We use an LP rounding framework that easily extends to other objectives such as maximizing volume of trade or profit. We justify our models by considering a dynamic marketplace setting where agents arrive according to a stochastic process and have finite patience (or deadlines) for being matched. We perform queueing analysis to show that for policies that route agents to facilities and match them, ensuring a low abandonment probability of agents reduces to ensuring sufficient flow arrives at each facility.

Motivation & Objective

• To model and solve the joint problem of pricing, scheduling, and facility placement in two-sided online marketplaces.
• To ensure high match quality via a bounded distance constraint between agents and facilities.
• To guarantee service availability by enforcing a minimum flow (lower bound L) at each facility.
• To maximize social surplus (total buyer value minus total seller cost) under weak budget balance (non-negative profit).
• To extend the framework to objectives such as maximizing profit or volume of trade.

Proposed method

• Formulates a new class of facility location problems—Two-sided Facility Location—where agents are routed to facilities based on metric distance.
• Uses an LP relaxation to model the problem, incorporating distance bounds R and minimum flow constraints L.
• Applies a novel LP rounding framework that achieves a (1+ε) approximation to the surplus objective for any ε>0.
• Relaxes the match quality constraint (maximum distance) by a constant factor to enable the approximation.
• Justifies the model via queueing theory, showing that low abandonment probability in dynamic markets reduces to sufficient flow at facilities.
• Reduces dynamic scheduling policies to the facility location problem by setting facility weights and lower bounds based on agent patience and arrival rates.

Experimental results

Research questions

• RQ1 How can a marketplace platform jointly optimize pricing, facility placement, and agent routing to maximize social surplus?
• RQ2 What approximation guarantees can be achieved for surplus maximization under distance and flow constraints?
• RQ3 How does incorporating agent patience (deadlines) affect facility location and matching performance?
• RQ4 Can facility location models be extended to capture truthful elicitation of agent deadlines via lotteries over prices and wages?
• RQ5 What is the relationship between facility flow, abandonment probability, and system performance in dynamic marketplaces?

Key findings

• The proposed LP rounding framework achieves a (1+ε) approximation to the social surplus for any ε>0, with a constant-factor relaxation in the match distance constraint.
• The framework extends naturally to objectives such as maximizing profit or volume of trade, maintaining the same approximation guarantee.
• Queueing analysis shows that to keep abandonment probability below η, the arrival rate λ must satisfy λ(�D + �S) ≥ 2/η, where �D and �S are average buyer and seller deadlines.
• The facility location model with lower bound L = 2/η captures the dynamic marketplace policy, improving the lower bound dependence from 1/η² to 1/η compared to patience-ignorant models.
• The model justifies the use of facility-based routing as a proxy for optimal scheduling policies in dynamic, stochastic marketplaces with deadline-aware agents.
• The results suggest that even partial knowledge of agent patience significantly improves system performance, enabling higher match quality with lower abandonment.