Stackelberg Dynamic Location Planning under Cumulative Demand

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.

💡 Research Summary

This paper tackles a dynamic facility location problem that simultaneously incorporates two realistic features often omitted in the literature: market competition and cumulative demand. Unlike the standard monopoly assumption, the authors model a duopolistic setting where a leader firm makes location decisions first and a follower firm reacts in each planning period. Unmet customer demand from one period carries over to the next, creating a feedback loop between past location choices and future demand. The problem is formulated as a Stackelberg game and expressed as two bilevel mixed‑integer programming (MIP) models: an optimistic variant, where the follower selects the response most favorable to the leader, and a pessimistic variant, where the follower chooses the least favorable response.

The authors prove that the optimistic variant is Σ²ᵖ‑hard by a polynomial‑time reduction from a Σ²ᵖ‑complete quantified Boolean formula, establishing that the problem lies at a higher level of the polynomial hierarchy than ordinary NP‑hard problems. Consequently, generic bilevel solvers that rely on KKT reformulations or simple branch‑and‑bound are inadequate for realistic instance sizes.

To overcome this computational barrier, the paper introduces a tailored branch‑and‑cut algorithm that repeatedly injects value‑function cuts into the leader’s master problem. After fixing a candidate facility schedule, the follower’s lower‑level problem (a mixed integer program that decides its own facilities and allocates demand) is solved. The optimal objective value and dual information are then used to construct tight upper‑ and lower‑bound cuts on the leader’s objective. The cuts are further strengthened by leveraging Lagrangian relaxations and subgradient information, dramatically shrinking the search tree.

Computational experiments on a benchmark set of 200 randomly generated instances (varying the number of periods, candidate sites, and facility caps) demonstrate that the proposed algorithm outperforms state‑of‑the‑art general‑purpose bilevel solvers by an average factor of 12 and up to 45 in worst‑case runtime, while maintaining optimality gaps below 0.1 %.

From a managerial perspective, the study quantifies the “cost of monopoly” – assuming a single firm controls the market leads to an average profit overestimation of about 50 % compared with the competitive Stackelberg outcome. Conversely, when the two firms cooperate and jointly optimize locations, total joint profit rises by roughly 6 % relative to the non‑cooperative equilibrium. Sensitivity analysis reveals that the adverse impact of competition intensifies as the demand‑carry‑over rate increases, highlighting the importance of coordination in markets with high demand persistence. The paper also provides guidelines on how parameters such as carry‑over rate, facility installation cost, and market size shape the robustness of location schedules, enabling practitioners to decide between conservative (risk‑averse) and aggressive (profit‑maximizing) strategies.

In summary, the work makes three major contributions: (1) it introduces a novel dynamic Stackelberg location model that captures cumulative demand; (2) it establishes Σ²ᵖ‑hardness for the optimistic variant and delivers a powerful branch‑and‑cut algorithm with value‑function cuts; and (3) it delivers extensive computational evidence and managerial insights that underscore the pitfalls of monopoly assumptions and the modest gains achievable through cooperation.