[Paper Review] Exact solutions to a carsharing pricing and relocation problem under uncertainty

Giovanni Pantuso|arXiv (Cornell University)|Mar 22, 2022

Transportation and Mobility Innovations4 citations

TL;DR

This paper proposes an exact solution method for a two-stage stochastic mixed-integer program that jointly optimizes carsharing pricing and vehicle relocation under demand uncertainty. Using an integer L-Shaped method with exact subproblem resolution, the approach outperforms commercial solvers on moderately sized Milan-based instances, showing that dynamic pricing significantly boosts demand and reduces reliance on staff-based relocations when fleet size is sufficient.

ABSTRACT

In this article we study the problem of jointly deciding carsharing prices and vehicle relocations. We consider carsharing services operating in the context of multi-modal urban transportation systems. Pricing decisions take into account the availability of alternative transport modes, and customer preferences with respect to these. In order to account for the inherent uncertainty in customer preferences, the problem is formulated as a mixed-integer two-stage stochastic program with integer decision variables at both stages. We propose an exact solution method for the problem based on the integer L-Shaped method which exploits an efficient exact algorithm for the solution of the subproblems. Tests on artificial instances based on the city of Milan illustrate that the method can solve, or find good solutions to, moderately sized instances for which a commercial solver fails. Furthermore, our results suggest that, by adjusting prices between different zones of the city, the operator can attract significantly more demand than with a fixed pricing scheme and that such a pricing scheme, coupled with a sufficiently large fleet, significantly reduces the relevance of staff-based relocations. A number of issues, that remain to be addressed in future research, are pointed out in our conclusions.

Motivation & Objective

• To address the joint optimization of dynamic pricing and vehicle relocation in one-way carsharing systems under uncertain customer preferences.
• To model the problem as a two-stage stochastic mixed-integer program with integer variables in both stages, capturing uncertainty in demand and user behavior.
• To develop an exact solution method capable of solving moderately sized instances where commercial solvers fail.
• To evaluate the impact of dynamic pricing and fleet size on demand capture and staff relocation requirements.

Proposed method

• Formulate the problem as a mixed-integer two-stage stochastic program with integer variables in both the first (pricing and relocation decisions) and second (demand realization) stages.
• Apply the integer L-Shaped method to decompose the problem and solve the master problem with Benders-type cuts.
• Design an efficient exact algorithm to solve the subproblems arising in each iteration of the L-Shaped method.
• Use scenario-based demand realizations to model uncertainty in customer preferences across different city zones.
• Incorporate collective dynamic pricing—adjusting prices per zone—to influence cumulative demand distribution.
• Test the method on artificial instances derived from Milan’s urban layout to evaluate performance and policy insights.

Experimental results

Research questions

• RQ1 How does dynamic, zone-specific pricing affect overall demand capture compared to fixed pricing in a one-way carsharing system?
• RQ2 To what extent can dynamic pricing reduce the need for staff-based vehicle relocations?
• RQ3 Can an exact solution method based on the integer L-Shaped approach solve moderately sized instances where commercial solvers fail?
• RQ4 How does fleet size influence the effectiveness of dynamic pricing in reducing relocation needs?
• RQ5 What is the trade-off between solution quality (optimality gap) and computational time when using the proposed method versus heuristic alternatives?

Key findings

• The proposed integer L-Shaped method successfully solves or finds high-quality solutions to moderately sized instances where commercial solvers fail, particularly for larger problem sizes.
• Dynamic pricing that varies by zone increases demand capture significantly compared to fixed pricing, with optimality gaps in the 0.0000–0.0963 range for small instances.
• When fleet size is sufficiently large, dynamic pricing alone can reduce the relevance of staff-based relocations, with optimality gaps increasing only moderately (up to 456.48%) for large instances.
• For instances with identical customer profiles and high fleet availability (α^V = 0.8), the method achieves near-optimal solutions (e.g., 0.0000% gap) in under 27 seconds for small instances.
• The solution time for the integer L-Shaped method remains stable and often faster than heuristic alternatives like ILS, especially in small-to-medium instances with low uncertainty.
• The method demonstrates robustness across varying demand patterns (α^FROM and α^TO), with consistent performance even under high demand imbalance scenarios.