[Paper Review] Distributionally Robust Facility Location Problem under Decision-dependent Stochastic Demand

Beste Basciftci, Shabbir Ahmed+1 more|arXiv (Cornell University)|Dec 11, 2019

Facility Location and Emergency ManagementBusiness, Management and Accounting43 references21 citations

TL;DR

This paper proposes a distributionally robust facility location model where customer demand distributions depend on facility location decisions, using piecewise linear functions to represent moment-based demand uncertainty. By leveraging duality and convex envelopes, the authors derive an exact mixed-integer linear programming reformulation and validate it through extensive computational studies, showing significant improvements in profit and service quality over traditional stochastic and robust models, with up to 19% speed-up via derived valid inequalities.

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.

Motivation & Objective

• To address the strategic facility location problem under demand uncertainty, where location decisions influence customer demand behavior.
• To model moment-based ambiguity sets for stochastic demand that are explicitly dependent on facility location decisions.
• To develop an exact mixed-integer linear programming reformulation of the decision-dependent distributionally robust model using duality and convex envelopes.
• To strengthen the formulation via derived valid inequalities and evaluate computational efficiency and solution quality.
• To demonstrate superior performance in profit, unmet demand reduction, and robustness compared to existing stochastic and robust optimization approaches.

Proposed method

• Demand mean and variance at each customer site are modeled as piecewise linear functions of facility location decisions, capturing the impact of location on customer behavior.
• A distributionally robust optimization framework is formulated using a moment-based ambiguity set that depends on location decisions.
• The inner problem is reformulated using linear programming duality and convex envelopes to produce an exact mixed-integer linear programming (MILP) representation.
• Valid inequalities are derived and incorporated to strengthen the formulation and improve computational performance.
• The model is solved via standard MILP solvers, with computational experiments comparing it to stochastic programming (SP) and distributionally robust (DR) models under decision-independent assumptions.
• Computational studies use varying instance sizes and parameter settings, including different ρ-values in the ρ-means approach to model varying degrees of location dependency.

Experimental results

Research questions

• RQ1 How does incorporating decision-dependent demand uncertainty improve facility location decisions compared to traditional stochastic and robust models?
• RQ2 To what extent do location decisions influence customer demand, and how can this dependency be modeled using moment-based distributions?
• RQ3 Can an exact mixed-integer linear programming reformulation be derived for a decision-dependent distributionally robust facility location problem?
• RQ4 How effective are the proposed valid inequalities in improving computational efficiency and solution speed?
• RQ5 What are the trade-offs between solution quality (profit, unmet demand) and computational time in the proposed framework?

Key findings

• The proposed decision-dependent distributionally robust model achieves significantly higher profit and lower unmet demand compared to both stochastic programming and standard distributionally robust models.
• The model demonstrates robust performance across diverse problem characteristics, including varying instance sizes and demand dependency patterns.
• The inclusion of valid inequalities leads to computational speed-ups of 3% to 19% across different instance sizes, with the largest improvement observed in larger instances.
• The distance-based approach opens most facilities due to proximity, while ρ-means approaches show shifting facility selections as ρ increases, reflecting reduced sensitivity to distance and increased influence of cost factors.
• The DDDR model’s run time increases more rapidly with instance size than SP or DR, but the performance gains justify the computational cost.
• The formulation’s exactness is validated through duality and convex envelope techniques, ensuring no optimality gap in the MILP reformulation.