Distributionally Robust Facility Location Problem under Decision-dependent Stochastic Demand (1912.05577v2)

Abstract: Facility location decisions significantly impact customer behavior and consequently the resulting demand in a wide range of businesses. Furthermore, sequentially realized uncertain demand enforces strategically determining locations under partial information. To address these issues, we study a facility location problem where the distribution of customer demand is dependent on location decisions. We represent moment information of stochastic demand as a piecewise linear function of facility-location decisions. Then, we propose a decision-dependent distributionally robust optimization model, and develop its exact mixed-integer linear programming reformulation. We further derive valid inequalities to strengthen the formulation. We conduct an extensive computational study, in which we compare our model with the existing (decision-independent) stochastic and robust models. Our results demonstrate superior performance of the proposed approach with remarkable improvement in profit and quality of service by extensively testing problem characteristics, in addition to computational speed-ups due to the formulation enhancements. These results draw attention to the need of considering the impact of location decisions on customer demand within this strategic-level planning problem.

Distributionally Robust Facility Location Problem under Decision-dependent Stochastic Demand: A Synopsis

The paper focuses on a nuanced exploration of facility location problems in contexts where customer demand is both uncertain and influenced by those very location decisions. This intricate dynamic is addressed by employing a decision-dependent distributionally robust optimization framework, which marks a significant departure from traditional models that treat demand distribution as independent from the location choices.

Key Contributions and Methodology

The authors present a systematic approach encapsulating three critical advancements:

1. Demand Representation via Piecewise Linear Functions: A foundational aspect of this paper is the representation of customer demand's moment information as a piecewise linear function of location decisions. This model accounts for the fact that opening new facilities can stimulate demand, linking demand directly to strategic location choices.
2. Model Formulation and Strengthening: The paper details the formulation of a decision-dependent distributionally robust facility location model, employing mixed-integer linear programming (MILP) techniques to enable an exact solution. Through duality and convex envelopes, the authors reformulate the problem, deriving valid inequalities to fortify the model further.
3. Computational Evidence and Benchmarking: An extensive computational paper substantiates the proposed approach's effectiveness, comparing it against decision-independent stochastic and robust optimization models. The results indicate a pronounced improvement in profitability and service quality, emphasizing the need for incorporating decision-dependent demand considerations in strategic facility planning.

Numerical and Comparative Results

The computational findings underscore the proposed model's superiority, demonstrating substantial enhancements in both objective values (profitability) and unmet demand levels. When juxtaposed with traditional stochastic programming (SP) and robust (DR) methods, the decision-dependent model consistently delivers superior outcomes, particularly in scenarios of high demand variability. These results remain robust, even under alternative demand distribution assumptions such as Gamma, underscoring the approach's adaptability across different business contexts.

Practical and Theoretical Implications

The implications of this research extend across both theoretical and practical domains. By challenging the conventional assumption of demand distribution independence, it opens new avenues for optimization under uncertainty, especially in sectors like carsharing and supply chain management where location-induced demand variability is pronounced. The MILP-based tractable reformulation and the computational speed-ups achieved through valid inequalities contribute to its practical viability, making it more accessible for deployment in real-world applications.

Future Directions

While the paper primarily focuses on facility locations in the logistics and transportation domain, the devised methodology could inspire application and further research in other sectors involving strategic asset allocation under uncertainty. Future research might probe deeper into alternative ambiguity sets or explore decision-dependent uncertainty's broader implications across varying strategic- and operational-level decisions in different industries.

In conclusion, the paper provides a meticulously crafted framework for addressing a complex aspect of facility location planning, challenging preconceived notions about demand independence and deploying advanced optimization techniques to derive insightful, robust solutions. This investigatorial endeavor not only broadens the scope of distributionally robust optimization literature but also offers practical solutions that could redefine operational strategies in the face of uncertainty.