Dynamic facility location problems predominantly suppose a monopoly over the service or product provided. Nonetheless, this premise can be a severe oversimplification in the presence of market competitors, as customers may prefer facilities installed by one of them. The monopolistic assumption can particularly worsen planning performance when demand depends on prior location decisions of the market participants, namely, when unmet demand from one period carries over to the next. Such a demand behaviour creates an intrinsic relationship between customer demand and location decisions of all market participants, and requires the decision-maker to anticipate the competitor's response. This work studies a novel competitive facility location problem that combines cumulative demand and market competition to devise high-quality solutions. We propose bilevel mixed-integer programming formulations for two variants of our problem, prove that the optimistic variant is $Σ^{p}_{2}$-hard, and develop branch-and-cut algorithms with tightened value-function cuts that significantly outperform general-purpose bilevel solvers. Our results quantify the severe cost of planning under a monopolistic assumption (profit drops by half on average) and the gains from cooperation over competition (6% more joint profit), while drawing managerial guidelines on how instance attributes and duopolistic modelling choices shape robust location schedules.
Dynamic Facility Location Problems (FLPs) are at the heart of strategic production and supply chain planning (Nickel and Saldanha-da Gama, 2019). Location decisions typically involve costly site-specific long-term commitments (e.g., to acquire properties and build infrastructure), making relocation or closure during the planning horizon undesirable. Throughout the last decade, several economic shifts have contributed to the emergence of temporary facilities (see, e.g., Rudkowski et al., 2020;Rosenbaum et al., 2021;Cao et al., 2024), which remain open at some locations for a short period of time (e.g., weeks or months) before moving to other locations. For example, in retail, temporary pop-up stores have been integrated into operations for many years and are expected to exceed 95 billion dollars in revenue in 2025 (Capital One Shopping, 2025). By serving different regions at a time, companies may create a sense of urgency among customers currently covered by a facility, while leading to an increase of interest (hype) and demand among those left uncovered.
From a mathematical optimization point of view, the planning for temporary facility location exhibits several unique characteristics. Classical mathematical formulations for dynamic FLPs assume that customer demand is independently defined for each time period of the planning horizon (see, e.g., Jena et al., 2015;Marín et al., 2018;Alizadeh et al., 2021;Vatsa and Jayaswal, 2021). However, for several commodities, this may severely misrepresent the real underlying demand behaviour. Indeed, unmet demand for non-perishable items such as personal clothing, beauty products, tech gadgets, and handmade crafts may not simply vanish, but is likely to persist in the future. Temporary facilities may therefore cause a more intrinsic relationship between location decisions and customer demand (Silva et al., 2025), as demand may carry over to future time periods and build up while facilities are not accessible. Such demand behaviour has recently been addressed in the literature, referred to as Cumulative Customer Demand (CCD) (see, e.g., Daneshvar et al., 2023;Silva et al., 2025), often exhibiting drastically different structures of optimal solutions that favour the relocation of facilities over the planning horizon.
Indeed, when companies operate in a monopoly, they may intentionally decide to let customer demand build up over time while operating in different regions, maximizing the total profit extracted over the planning horizon (Silva et al., 2025). However, such a strategy is considerably vulnerable when competing firms are present, as the competition may capture the accumulated customer demand before the company. Explicitly accounting for competition within the planning stage is therefore of crucial importance to ensure profitability in such dynamic and competitive markets. The literature on competitive FLPs (see, e.g. Ljubić and Moreno, 2018;Beresnev and Melnikov, 2019;Qi et al., 2022) has modelled market competition mostly as a Stackelberg game (Von Stackelberg, 1934). However, to the best of our knowledge, existing works in this research stream are restricted to a single time period, and therefore cannot readily account for cumulative customer demand.
In this work, we investigate how to handle market competition under cumulative demand. We consider a company, further referred to as the leader, that must place her temporary facilities over the planning horizon to capture customers and maximize her profit. The leader is aware of a competitor, further referred to as the follower, that will react by locating his temporary facilities over the planning horizon to capture customers and maximize his profit. Customers patronize facilities installed by the leader or the follower at each period based on their preferences, and unmet demand carries over to subsequent periods. This planning problem, referred to as the Competitive Dynamic Facility Location Problem under Cumulative Customer Demand (CDFLP-CCD), combines cumulative demand and market competition to produce high-quality location schedules for the leader. As such, we contribute to the literature on location problems as follows:
We introduce the CDFLP-CCD, a new temporary facility location problem that requires to merge distinctive features which previously have been considered separately. We examine both the optimistic variant of the respective Stackelberh competition, which is more commonly studied in the literature, and the considerably less explored pessimistic variant of the CDFLP-CCD.
We explicitly prove the Σ p 2 -hardness of the optimistic variant, even if each player is limited to locating a single facility over the planning horizon. To the best of our knowledge, we present the first Σ p 2 -hardness result for a non min-max, non zero-sum competitive location problem (i.e., each player maximizes its profit, and the loss of a player does not imply the gain of another). We thus enrich the catalogue
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