Algorithms for min-buying in networks

TL;DR

This work addresses min-buying pricing on networks by formulating a bipartite market of outlets and demand routes, and by introducing two pricing models: MNPP with deterministic allocation and BMNPP with binary-logit buyer choice. It develops two MIP formulations (MNPP-IP1 and MNPP-IP2) and a suite of selection and insertion heuristics to solve MNPP and BMNPP under a price-ladder discrete price set, comparing performance via a detailed numerical study. Key findings show price matching can substantially boost revenue when brand sensitivity is estimable (BMNPP), while price wars dominate when it is not, and network structure significantly shapes pricing outcomes; overall, the proposed MFIP/heuristic toolkit delivers meaningful gains over a simple single-price baseline. The results offer practical guidance for forecourt pricing and lay the groundwork for industrial-scale, hyper-heuristic solutions and extensions to evolving network contexts such as EV charging and strategic outlet design.

Abstract

The paper is motivated by pricing decisions faced by forecourt fuel retailers across their outlets on a road network. Through our modelling approach we are able adapt the network structure to a bipartite graph with demand nodes representing volumes of fuel from customers using a specific route that connects to the seller's outlet nodes that intersect that route on the network. Customers may have their demand satisfied at the lowest priced competitor on their route. However, the seller can satisfy some or all of this demand by matching or beating this price via one of their outlets intersecting the route. We give a practical extension to min-pricing by considering a binary logit variant for buyers evaluating the choice between two sellers. We derive two MIP formulations for min-buying in the case of general demand. We also propose several constructive heuristics, based on insertion and selection operations, suitable for problem instances beyond the scope of the exact methods. The performance of models and algorithms are evaluated in a numerical study and develop insights from the results. Importantly, we are able to highlight the value of price-matching decisions under buyer demand sensitivity.

Figures (16)

• Figure 1: Boxplots summarising times taken by the exact and heuristic algorithms for MNPP.
• Figure 2: Boxplots summarising percentage optimality gaps and gains over SP for MNPP.
• Figure 3: Boxplots summarising times taken by the FI and orderI by $|L|$ for MNPP.
• Figure 4: Run times of FI by $|\mathcal{O}|$, $|\mathcal{N}|$ and $P$
• Figure 5: Run times of orderI by $|\mathcal{O}|$, $|\mathcal{N}|$ and $P$

Theorems & Definitions (1)

• Remark 2