[Paper Review] Maximum Customers' Satisfaction in One-way Car-sharing: Modeling, Exact and Heuristic Solving

Welverton R. Silva, Rafael C. S. Schouery|arXiv (Cornell University)|Aug 13, 2020

Transportation and Mobility Innovations4 citations

TL;DR

This paper proposes exact and heuristic methods to maximize customer satisfaction in one-way car-sharing systems with two stations and known, bidirectional trip demands. It introduces a mixed-integer programming model with strengthened constraints and matheuristics based on local search (GRASP, VNS, TS), achieving optimality for 85% of 1,000-customer instances within 10 minutes and outperforming MIP on larger instances with gaps under 11% on average.

ABSTRACT

One-way car-sharing systems are transportation systems that allow customers to rent cars at stations scattered around the city, use them for a short journey, and return them at any station. The maximum customers' satisfaction problem concerns the task of assigning the cars, initially located at given stations, to maximize the number of satisfied customers. We consider the problem with two stations where each customer has exactly two demands in opposite directions between both stations, and a customer is satisfied only if both their demands are fulfilled. For solving this problem, we propose mixed-integer programming (MIP) models and matheuristics based on local search. We created a benchmark of instances used to test the exact and heuristic approaches. Additionally, we proposed a preprocessing procedure to reduce the size of the instance. Our MIP models can solve to optimality 85% of the proposed instances with 1000 customers in 10 minutes, with an average gap smaller than 0.1% for all these instances. For larger instances (2500 and 5000 customers), except for some particular cases, they presented an average gap smaller than 0.8%. Also, our local-based matheuristics presented small average gaps which are better than the MIP models in some larger instances.

Motivation & Objective

• To address the maximum customers’ satisfaction problem in one-way car-sharing systems with two stations and known, bidirectional trip demands.
• To design exact and heuristic algorithms that maximize the number of satisfied customers, where satisfaction requires fulfilling both a pickup and drop-off demand.
• To develop a preprocessing procedure to reduce instance size and improve computational efficiency.
• To evaluate the performance of MIP models and matheuristics on a new benchmark of synthetic instances.
• To demonstrate the scalability and robustness of the proposed methods on large-scale instances.

Proposed method

• Formulates the problem as a mixed-integer programming (MIP) model based on network flow, with two variants: (CS1) original model and (CS2) a strengthened version with additional constraints to improve bounds.
• Proposes a preprocessing procedure that reduces instance size by eliminating redundant demands and simplifying time intervals, improving both exact and heuristic performance.
• Develops three matheuristics—GRASP, VNS, and tabu search—using a greedy initial solution derived from linear programming relaxation.
• Employs local search neighborhoods to iteratively improve solutions, with diversification mechanisms to escape local optima.
• Uses a benchmark of 120 instances (100–5000 customers) with varying time window distributions and fleet sizes to evaluate performance.
• Implements a time limit of 10 minutes for all exact and heuristic runs, with solution quality measured by optimality gap and number of optimal solutions.

Experimental results

Research questions

• RQ1 Can a strengthened MIP formulation significantly improve solution quality and convergence speed for the maximum customers’ satisfaction problem in two-station one-way car-sharing?
• RQ2 How effective is the proposed preprocessing procedure in reducing instance size and enhancing the performance of both exact and heuristic methods?
• RQ3 Do matheuristics based on local search outperform MIP models on large-scale instances where MIP struggles due to time limits?
• RQ4 What is the trade-off between solution quality and computation time when using GRASP, VNS, and tabu search in this context?
• RQ5 Can the proposed methods be generalized to larger systems with more than two stations and more complex customer demand patterns?

Key findings

• The MIP model (CS2) achieved optimality for 85% of 1,000-customer instances within 10 minutes, with an average optimality gap of less than 0.1%.
• For 2,500 and 5,000 customer instances, the MIP models maintained an average gap below 0.8%, except in a few difficult cases.
• The matheuristic GRASP outperformed both VNS and tabu search, achieving an average gap of less than 2% on 1,000-customer instances.
• On 5,000-customer instances, the MIP models had an average gap exceeding 39%, while the best matheuristics achieved a gap of less than 11%.
• The preprocessing procedure reduced instance size effectively and performed comparably to commercial MIP presolve, with added benefit for heuristic design.
• The local search phase consumed most of the 10-minute time limit in large instances, indicating high computational cost despite good initial solutions.