Robust Multi-Objective Optimization for Bicycle Rebalancing in Shared Mobility Systems

Diego Daniel Pedroza-Perez; Gabriel Luque; Sergio Nesmachnow; Jamal Toutouh

cs.NEApr 9, 2026

Robust Multi-Objective Optimization for Bicycle Rebalancing in Shared Mobility Systems

Authors: Diego Daniel Pedroza-Perez, Gabriel Luque, Sergio Nesmachnow, Jamal Toutouh

Organizations: ITIS Software, University of Malaga, Spain · Universidad de la República, Uruguay

Abstract

Dock-based bike-sharing systems exhibit spatial imbalances between bicycle supply and user demand, often addressed through overnight truck-based rebalancing. This work studies static overnight rebalancing under demand uncertainty modeled as a tri-objective optimization problem. The objectives minimize total travel distance, expected unmet demand, and a robustness-oriented unmet demand measure over high-demand scenarios. Route plans are evaluated via a recourse simulation that enforces truck loads and station capacity constraints across multiple demand realizations. The robustness objective supports selecting plans that reduce peak-demand service degradation. Trade-off solutions are approximated with Non-dominated Sorting Genetic Algorithm II using a permutation--partition encoding and domain-specific relocation operators, including a biased best-improvement move for station relocation. Experiments on the real Barcelona Bicing system with 460 stations show well-distributed Pareto sets and substantial contributions to the reference non-dominated set. Greedy constructive baselines mainly yield extreme solutions and are often dominated.

Figures & tables

Figure 1. Example of solution encoding of a scenario with 9 stations and 3 trucks.

Figure 2. Diagram showing the operation of the AB2 operator. The red arrows indicate the step sequence.

Figure 3. BB1-MAX operator diagram. The red arrows indicate the step sequence.

Figure 4. Diagram of the Station Rebalancing using Roulette-Wheel Selection operator (BB2-MIN/MAX).

Figure 5. Instance map: the red points are the bike-shared stations and the blue is the depot.

Parameter	Domain
Crossover on $\mathcal{\mathbf{R}}$ ( $C_{R}$ )	OX, PMX, ERX, EERX
Mutation on $\mathcal{\mathbf{R}}$ ( $M_{R}$ )	IM, BMM, BSM, SM
Probability of mutation $\mathcal{\mathbf{R}}$ ( $PM_{R}$ )	{0.01, 0.05, 0.1, 0.15, 0.20, 0.25}
Probability of crossover $\mathcal{\mathbf{R}}$ ( $PC_{R}$ )	{0.1, 0.25, 0.5, 0.7, 0.9, 1}
Probability of mutation $\boldsymbol{\vec{t}}$ ( $PM_{T}$ )	{0.01, 0.05, 0.1, 0.15, 0.20, 0.25}

Table 1. Candidates for the hyperparameters

Operator	$C_{R}$	$M_{R}$	$PM_{R}$	$PC_{R}$	$PM_{T}$
AB2	PMX	SM	0.25	0.1	0.15
BB1-MIN	PMX	SM	0.25	0.1	0.20
BB1-MAX	PMX	BSM	0.01	0.1	0.25
BB2-MAX	PMX	SM	0.20	0.1	0.25
BB2-MIN	PMX	SM	0.20	0.1	0.15

Table 2. Parameter configurations for each operator.

	Min	Median	IQR	Max
AB2	0.445	0.504	0.035	0.533
BB1-MAX	0.620	0.750	0.054	0.851
BB1-MIN	0.360	0.438	0.044	0.530
BB2-MAX	0.388	0.503	0.040	0.551
BB2-MIN	0.382	0.467	0.039	0.541

Table 3. Results of rhv for the studied instance.

	IGD+				GD+
	Min	Med	IQR	Max	Min	Med	IQR	Max
AB2	0.156	0.181	0.014	0.210	0.110	0.141	0.029	0.186
BB1-MAX	0.043	0.065	0.009	0.093	0.062	0.078	0.006	0.102
BB1-MIN	0.179	0.217	0.027	0.252	0.114	0.159	0.025	0.200
BB2-MAX	0.144	0.176	0.016	0.208	0.116	0.151	0.027	0.183
BB2-MIN	0.168	0.203	0.023	0.249	0.106	0.152	0.029	0.199

Table 4. Results for IGD+ and GD+ metrics.

	Spread				#NDS
	Min	Med	IQR	Max	Min	Med	IQR	Max
AB2	5.074	5.680	0.481	6.142	194	198	2.25	200
BB1-MAX	2.247	2.576	0.456	4.126	188	199	2.00	200
BB1-MIN	5.583	6.462	0.390	6.991	192	198	2.00	200
BB2-MAX	4.911	5.348	0.346	5.904	190	198	1.25	200
BB2-MIN	5.121	5.981	0.480	6.642	190	198	2.00	200

Table 5. Results for Spread and #NDS metrics.

Figure 6. Three-dimensional Pareto front obtained by the different mutation operators.

Figure 7. 2D front projection: distance vs. critical unmet demand.

	rhv	gd+	igd+	spread	#nds
BB1-MAX	0.975	0.004	0.007	1.241	36
BB1-MIN	0.681	0.092	0.153	1.564	1
BB2-MAX	0.742	0.062	0.111	2.046	3
BB2-MIN	0.696	0.087	0.142	1.699	3
GLOBE_SD	0.402	0.007	0.127	0.414	2
GLOBE_Dist	0.251	0.048	0.180	0.377	0

Table 6. Results of MO metrics.

Figure 8. Two-dimensional Reference Pareto front projection: distance vs. critical unmet demand.

Figure 9. Histogram of the number of bikes needed per station in the non-rebalancing configuration (BASE), compared with the closest to ideal vector in normal and critical cases.

Figure 10. Total distance distribution of the solutions of our approach by number of trucks.

Figure 11. Unmet demand distribution of the solutions of our approach by number of trucks.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 12. Heatmap of pairwise comparisons based on the rhv metric.

Figure 13. Heatmap of pairwise comparisons based on the gd+ metric.

Figure 14. Heatmap of pairwise comparisons based on the igd+ metric.

Figure 15. Heatmap of pairwise comparisons based on the spread metric.

Figure 16. Heatmap of pairwise comparisons based on the #nds metric.

Multi-Objective Optimization

Multi-objective optimization studies how to find and choose solutions that balance competing goals, constraints, and preferences.

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This paper proposes a fully dynamic Deep Reinforcement Learning (DRL) method for rebalancing dockless bike-sharing systems, overcoming the limitations of periodic, system-wide interventions. We model the service through a graph-based simulator and cast rebalancing as a Markov decision process. A DRL agent routes a single truck in real time, executing localized pick-up, drop-off, and charging actions guided by spatiotemporal criticality scores. Experiments on real-world data show significant reductions in availability failures with a minimal fleet size, while limiting spatial inequality and mobility deserts. Our approach demonstrates the value of learning-based rebalancing for efficient and reliable shared micromobility.

Abstract

Multi-Objective Optimization

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