Stackelberg Dynamic Location Planning under Cumulative Demand

TL;DR

The study tackles a dynamic, competitive facility location problem under cumulative demand, formulating the Competitive Dynamic Facility Location Problem under Cumulative Customer Demand (CDFLP-CCD) as a bilevel mixed-integer program for optimistic and pessimistic follower behavior. It proves the optimistic variant is $Σ^{p}_{2}$-hard and introduces specialized branch-and-cut methods with tightened value-function cuts that outperform general solvers, achieving substantial improvements in solving time and optimality gaps. Computational experiments on Quebec-based synthetic benchmarks reveal that accounting for competition can drastically improve outcomes over monopoly assumptions (about a 52% average opportunity gap when ignored) and that cooperation adds about 6% more joint profit on average. The work provides actionable managerial insights into how instance attributes, demand dynamics, and duopolistic modeling choices shape robust location schedules, with broader implications for dynamic, multi-period competitive planning problems.

Abstract

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.

Figures (17)

• Figure 1: Potential interferences of the leader on the profit of the follower under bilevel feasible solution $(\boldsymbol{y}^{\prime}, \boldsymbol{z}^{\star})$. Nodes represent periods, while arcs represent feasible transitions between periods, according to Constraints \ref{['eq:third-detailed-optimistic-ct1']}-\ref{['eq:third-detailed-optimistic-ct10']}.
• Figure 2: Runtime ratios for exact methods, where the $y$ axis presents the percentage of instances with a ratio smaller than or equal to the reference value on the $x$ axis. [Optimistic variant of the CDFLP-CCD]
• Figure 3: Objective ratios for exact methods, where the $y$ axis presents the percentage of instances with a ratio smaller than or equal to the reference value on the $x$ axis. [Optimistic variant of the CDFLP-CCD]
• Figure 4: Opportunity gaps under the monopolistic assumption (left) and prices of competition (right) grouped by different instance attributes, with the number of instances between square brackets. [Optimistic variant of the CDFLP-CCD]
• Figure 5: Location schedules of the illustrative instance and its variations, where $\Pi^{L}$ and $\Pi^{F}$ indicate the objective function values of the leader and the follower, respectively. [Optimistic variant of the CDFLP-CCD]

Theorems & Definitions (7)

• Theorem 1
• Proposition 1
• Lemma 1
• Lemma 2
• Theorem 2
• Corollary 1
• Corollary 2