Hierarchical Optimization via LLM-Guided Objective Evolution for Mobility-on-Demand Systems
Authors: Yi Zhang
Yushen Long
Yun Ni
Liping Huang
Xiaohong Wang
Jun Liu
Venue: NeurIPS 2025 poster ()
Keywords: Dynamic Hierarchical Optimization
LLM-Optimizer Interaction
Mobility-on-Demand Systems
Meta-Objective Design
Feasibility-Aware Adaptation
Primary area: optimization
TL;DR: We propose a training-free hybrid framework where LLMs generate high-level goals via prompt-based harmony search and optimizers enforce constraints in dynamic ride-hailing, outperforming RL, manual decomposition, and LLM-only baselines.

Abstract: Online ride-hailing platforms aim to deliver efficient mobility-on-demand services, often facing challenges in balancing dynamic and spatially heterogeneous supply and demand. Existing methods typically fall into two categories: reinforcement learning (RL) approaches, which suffer from data inefficiency, oversimplified modeling of real-world dynamics, and difficulty enforcing operational constraints; or decomposed online optimization methods, which rely on manually designed high-level objectives that lack awareness of low-level routing dynamics. To address this issue, we propose a novel hybrid framework that integrates large language model (LLM) with mathematical optimization in a dynamic hierarchical system: (1) it is training-free, removing the need for large-scale interaction data as in RL, and (2) it leverages LLM to bridge cognitive limitations caused by problem decomposition by adaptively generating high-level objectives. Within this framework, LLM serves as a meta-optimizer, producing semantic heuristics that guide a low-level optimizer responsible for constraint enforcement and real-time decision execution. These heuristics are refined through a closed-loop evolutionary process, driven by harmony search, which iteratively adapts the LLM prompts based on feasibility and performance feedback from the optimization layer. Extensive experiments based on scenarios derived from both the New York and Chicago taxi datasets demonstrate the effectiveness of our approach, achieving an average improvement of 16% compared to state-of-the-art baselines.

Hierarchical Optimization via LLM-Guided Objective Evolution for Mobility-on-Demand Systems
Yi Zhang'*
Yushen Long²
Xiaohong Wang' Agency for Science, Technology and Research, Singapore Morgan Stanley Asia Pte.
Yun Ni Liping Huang'
Jun Liu
Onto Innovation Inc.
School of computing and communications, Lancaster University, UK
(Zhang_ Yi, BhtangLLpping Wang_XK Xiaohong) (aahhnng)a-sstr. sa-star edu. sg Yushen. Baaheennoongnorggatttaalllleee com, Yun N...Ni&ee eileonnoinnovtiinn com j. 1.1iu81@lamcasten ac uk
Abstract
Online ride-hailing platforms aim deliver efficient mobility-on-demand services, often facing challenges in balancing dynamic and spatially heterogeneous supply and demand. Existing methods typically fall into two categories: reinforcement learning (RL) approaches, which suffer from data inefficiency, oversimplified modeling of real-world dynamics, and difficulty enforcing operational constraints; or decomposed online optimization methods, which rely on manually designed high- level objectives that lack awareness of low-level routing dynamics. To address this issue, we propose novel hybrid framework that integrates large language model (LLM) with mathematical optimization fraining-free, removing the need for large-scale interaction data as in RL, and (2) it leverages L.LM to bridge cognitive limitations caused by problem decomposition by adaptively generating high-level objectives. Within this framework, L.L. serves as meta-optimizer, producing semantic heuristics that guide a low-level optimizer responsible for constraint enforcement and real-time decision exccution. These heuristics are refined through closed-loop evolutionary process, driven by harmony search, which iteratively adapts the LLM prompts based on feasibility and performance feedback from the optimization layer. Extensive experiments based scenarios derived from both the New York and Chicago taxi datasets demonstrate the effectiveness of our approach, achieving an average improvement of 16% compared to ottte-of-the-art baselines.
dynamic hierarchical system: (1) it is
Introduction
Mobilit--on-demand platforms, such as ride-hailing services, have become critical urban transporta- tion infrastructures. to address unbalanced demand and supply by continuously executing core
This sequential decision-makee making involves solving complex combinatorial optimization problems under spatiotemporal constraints, ensuring timely and efficient service delivery in ever-evolving urban environments.
Docision-making processes: order dispatch and vehicle routing II 4
5 6
10 have been employed learn policies to address the problem. These approaches can capture long-term rewards by considering the future trajectories of drivers and passengers. However, RL methods often require extensive training data and interactions to learn effectively. Training can be unstable and enforcing hard constraints chal- lenging. In addition, learned policies can behave unpredictably outside of their training distribution.
Firstly, reinforcement learning methods
*Corresooning Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Secondly, two-stage optimization frameworks have been deployed to decompose the problem to reduce the complexity. Typically, the high-level stage addresses supply-deman balancing, e.g.. batch matching via mixed-integer linear programming (MILP) 22
3 12, while the low-level stage 14 However, high-level objectives are often manually designed without full knowledge of the low-level dynamics, potentially leading to suboptimal decisions. This disconnect can lead to misaligned objectives that do not fully reflect the true operational constraints, ultimately resulting in inefficiencies like increased waiting times or idle distances. Thirdly, large language models offer a new paradigm: their embedded priors on combinatorial reasoning and urban mobility can improve adaptability. However, current applica- I7 18 191. When applied to dynamic ride-hailing systems with real-time state transitions, these methods face two key limitations: (1) absence of iterative refinement aligned with evolving system states, and (2) lack of solver integration, resulting in proposals that may violate constraints degrade solution quality. To address the above issue, we present the first integration of L.M and mathematical optimization for dynamic sequen- tial sreiision-maiing systems. Specifically, we propose hybrid LLM-oppimizer framework that decomposes the problem hicrarchically. strategically embedding LLM only where human expertise bottlenecks exist: (I) LLM as Meta-Objective Designer: Dynamically evolves high-level objectives via prompt-based harmony search (20 2II guided by feasibility feedback from the optimization solver. (2) Optimizer as Constraint Enforcer: Solves low-level routing with mathematical rigor, ensuring real-time feasibility. (3) Heuristics as Prompt Evolver: Leverages harmony search algo- rithm to iteratively refine LLM prompts, guided by optimizer feedback to adaptively explore and converge toward effective meta-objectives. This framework is training-free, eliminating the need for extensive data or interaction required by RL.-based methods. Simultaneously, by leveraging L.LM to adaptively evolve the high-level objective, it mitigates the sub-optimalit, introduced by manual design in decomposed optimization, aligning high-level decisions more closely with downstream dynamics.
focuses on dispatching or routing, e.g., graph-based search [3
tions focus primarily on static optimization settings [TS] 16
Our framework hierarchically decomposes each decision step into two levels: The high-level module is responsible for assigning passengers to taxis based on real-time spatial configurations and anticipated supply-demand imbalances, while the low-level module solves the routing or visiting sequence problem for each taxi to minimize passenger waiting time under spatiotemporal constraints. To address the partial observability challenge (high-level model lack foresight into downstream routing dynamics) induced by the decomposition, we employ LLM as meta-heuristic designer, leveraging its implicit understanding of urban mobility patterns to adaptively refine high-level objectives. As captured in Figure LLM generates high-level assignment objectives that serve as semantic guides within the optimization loop. These objectives are embedded into a closed-loop evolutionary process, where each simulation epoch evaluates their fitness. The evolutionary mechanism is guided by harmony search algorithm, which iteratively refines the LLM prompt space to improve objective quality. This feedback- mechanism enables the LLM- -eMeeeerrteed heuristics to adapt and improve over time, combining the semantic richness of LLM with the structural robustness of traditional optimization. By integrating LLM as a semantic objective generator within hicrarchical optimization loop. framework achieves dynamic adaptability absent in static operation research (OR) formulations, meanwhile avoiding the data inefficiency of RL-based approaches.
We summarize our contributions as follows: 1) We propose a hybrid LLM + Optimizer paradigm, where LLM acts as meta-optimizer for evolving high-level objective through prompt-based evolution, while mathematical optimization solver guarantees constraint satisfaction and numerical rigor. To the best of our knowledge, this is the first work to leverage the capabilities of LLM combined with optimization solver for sequential dynamic decision maaiing systems. 2) We introduce a harmony search algorithm with three novel operators (random inference, heuristic improvement, and innovative generation) to iteratively refine LLM- LLM-generated objectives using feedback from the mathematical solver. 3) Extensive experiments on various scenarios derived from the New York and Chicago taxi datasets demonstrate the effectiveness of our approach, reducing passenger waiting time by approximately 16% over mate-of-theear baselines.
2 Related work
Reinforcement Learning in Ride-Hailing RL-bbasd approaches have been adopted, which enables the system to learn from historical data and optimize decision-making by considering the expected

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Search Block (Heuristic Evolution)
Prompts with population size obtained from different operators (first part)
WAY Random Inference
Peredgg
g
Exploration-
Exploitation based elite parent
Exploitation
based
Inmovaative C senerration
Scenario
Geospadal
First layer bluepint
WAY
WAR
Popuistion with (Each individual: Objectives
multiples staps Fitness
Packing Tagether
Evaluation Block
Dynamic System Block
Frtneso
current environment
Parse the response
taxi position.
Recorder
Evalustor
prompt)
Archived
Adaustive ob objection from L.M
Objective Results
pick-up drop-off
taxi
Rect/pmmmnt constraints
results
Multr shreate (9
Simulatorr
Optimbeer
TAinimice passenger dolay
Docuiin Time const constralmts
Objective
Low-level Optanizer
Uprimized dispatching routing results
Figure 1: Overall control flow framework. Search block: Uses harmony search algorithm to iteratively select and apply 3 prompt-refinement operators. Initial iterations prioritize heuristics from Operator 1; Dynamic System block: At each timestep, the LLM generates high-level objectives based on the refined prompt and simulator-reported states (e.g., driver locations, pending orders). These objectives guide a two-level optimizer: high-level dispatcher assigns orders, and a low-level router determines feasible visiting sequences. The optimizer's decisions are executed updating system states. This closed-loop process continues until the simulation horizon concludes; Evaluation block: Computes the fitness score from simulator trajectories and pairs with LLM-inferred objectives to update the harmony search population.
simulator,
22 including the deep RL-based approaches uses RI. to estimate the long-term value of each possible driver-order assignment, considering just immediate trip rewards but also future opportunities (like driver repositioning). 100 integrates behavior prediction and combinatorial optimization with deep double scalable network generate order-driver assignments in auto-regressive manner. However, RL-based approaches often require large number of interactions to learn effectively, and their training can be unstable and highly sensitive hyperparameters.
future trajectories of drivers and passengers 10
11
5 8
23
On-line Optimization in Order Dispatching Order-taxi matching problems in ride-hailing plat- forms are typically f'rmullated as MILP models or framed as bipartite graph matching problems 13 14 |4] Combinatorial optimization problems often face scalability challenges due to
2
12
the large decision variable space, leading to multi-stage decompositions. (2) proposed federated optimization framework integrating assignment blocks and routing engines with limited information exchange. [24] developed two MILP models for ride-hailing: flow-based high-level model for supply-demand balance and a routing-based low-level model to minimize travel time. However, hierarchical decomposition introduces sub-optimality by decoupling optimization stages: high-level objectives lack visibility into low-level operational dynamics, resulting myopic decisions that fail to preserve system wide optimality.

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LLM for Operation Research Problems Recent studies have explored the use of LLMs to automate the solution of OR problems, with approaches falling into two main categories: (1)
automating mathematical model formulation to reduce reliance on domain expertise 18 17 25
26 19 27
Approaches such as task allocation, few-shot learning, and chain-of-experts have been
proposed to guide programmatic LLM prompts [28] 29 30
32 (2) Directly querying L.MSs for algorithmic code to find feasible solutions to OR problems 15 33
17,
31
34
35 36]. FunSearch
[1 leverages the generative capabilities of LLMs to propose candidate programs, which are then rigorously evaluated to identify high-quality solutions. The EoH framework (16] encodes heuristic concepts natural language "thoughts", which are then translated into executable code by LLM. Additionally, it employs an evolutionary algorithm to iteratively refine and evolve the prompts. However, the best of our knowledge, no studies have yet combined LLM with optimization formulations in the context of dynamic decision-making systems.
3 Methodology
3.1 Dynamic Hierarchical Optimization Problem
We propose a novel problem formulation for ride-hailing system (Appendix A.66) with a formal analysis of the associated search space complexity (Appendix (A.5). which jointly motivate the decomposition of the original task into two tractable subproblems involving sequential decisions in task assignment and vehicle routing. These subproblems are integrated within hierarchical framework augmented by an LLM component.
First-level assignment problem As an assignment problem, goal extends beyond merely determining the next immediate passenger for an idle taxi. Instead, we formulate a comprehensive assignment across entire system snapshot, ensuring that each passenger is served and assigned to most one taxi. Omitting additional variables that capture taxi dynamics at the first level significantly reduces problem complexity. However, traditional approaches employ handcrafted objectives (left below), but these lack awareness of low-level routing dynamics may lead to suboptimal system performance:
ydua Σ 8D (TRopgy + TRppg= (Distance)
min Jist
min
& = LLM(S Hc-1)
s.t. ΣVm
lst
Σ p0 Tp - tg=l (Temporal)
LLM Redise
s.t. Σ 5 =1
P.
y"" 10
Ist
(Uilization)
ydr (0,1)
Notes: O", D": Passenger p origin, destination points; 5": Taxi v start position. TRyy Travel time + j; T": Passenger request time; ts=: Taxi u start time. yd" E {0, 1): Binary assignment variable.
These objective components can be linearly combined through weighted summation: Jist = = 3. + JJuut where fixed weights a, 3, y trade off distance (vehicle km), temporal alignment (estimated waiting time) and utilization fairness. While this myopic approach lacks visibility into low-level dynamics,
Ji
framework replaces handcrafted objectives with LLM-generade LLM(S,, H.-1), where S, encodes vehicle positions and pending requests, and Hz-1 captures historical congestion and assignments. By embedding latent urban dynamics into MILP objectives, the LLM bridges the gap between assignment and routing without explicit dynamic modeling.
&
Second-level sequencing problem After solving the first-level problem and assigning passengers to each taxi, the second-level problem can be solved independently for each taxi, eliminating index v. In the formulation below, Op and Dp represent the origin and destination of passenger p. while S and E denote the taxi's source and sink depots. is the estimated arrival time. The pickup service point set P consists of {p, }²)}. and the dropoff service point set D consists of ((p, D*)}-
[b] ld initialize location and arrival time assignments, where æıj is binary variable indicating link selection. AR, and DP; denote arrival time and depart time at position i. Constraints IT) enforce flow conservation, ensuring equal incoming and outgoing links. Constraints (gg- mm define system dynamics: (1g establish taxi arrival times at dropoff points as the sum lkk enforce the same for new pickups.
Constraints
[T
of pickup departure times and travel durations, while

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Departure times must not precede arrival times [Tp. [ and must larger than or equal to passenger request times (Im). The objective Tap minimizes total passenger waiting time.
minimize
J2nd
Σ
(DP (DP(moor)) -
(la)
P
Σ 88,
(Ib)
D 2 DP (p(p, TR
iEP
-M(1 I(p,O"))(p.Dn)
(Ih)
ARs =
(1c)
(((,,Dr) 2 (p,)) AR ,r") ( S DP (p,DDr) + TR DROF
(1i)
Σ T1E
(1d)
iEP
+ +M11 (p),Dr),(' ,0r)))
(1j)
Σ Tij
Σ
Tjg: Vje ED
(Ie)
A 2 DP (P, + DROd
iEP
PEP
M(1 (I..D)),0 (w))))))
(1k)
Σ Fji Σ
Tik: Vi EP
(1f)
JED
RED
(p) S DP (p,00)) + DR
Di All
(II)
+ M(1 ((p.O"%)(p.De))
(1g)
DP (P( 2 TM
(1m)
3.2
L.M--Optimizer Interaction Protocol
3.2.1 Scenario prompt setup
The scenario prompt Pace equips the LLM with contextual knowledge regarding the problem, the assigned role, and the specific task. The prompt structure is formalized through three nested layers: Pare = Pays @ Pgeo @ P model.. (1) Pays defines the LLM's role aligning outputs with solver-compatible templates; (2) Pgeo encodes geospatial semantics through Manhattan Chicago zone graphs and OD matrices; (3) Pmodel provides class blueprints enable variable-aware objective formulation. In contrast to Pasta. which merely specifies input formats or argument types, Pmodel ensures LLM-generated objectives respect both class structure and solver constraints. Further details are provided in Appendix A.9.I
adaptive objective designer,
3.2.2 Dynamic environment feedback
Since each simulation involves multiple decision decision-making -aaing steps, there enabling inference with an LLM, as captured in Figure 2 (1) a one-time query at the beginning of the test, where the generated objective remains fixed for all subsequent steps, (2) queries each step of the test. These correspond to two inference strategies. From a control perspective,
approaches to
Open-loop control
Close-foop control
Optimaar
Optimioe
Optimicer
Optinizer
Optinizer
Optinizor
Stort
Sinskatter
Simudater
Figure 2: Inference strategies in a single simulation run: (1) Open-loop control. where a one-time query occurs the beginning of the test. (2) Closed-loop control, where queries occur each step of the test.
the second approach, where multiple queries occur, incorporates real-time environment feedback from the simulator, allowing the LLM to generate updated objectives dynamically. This forms closed-loop control system, which is generally more effective for adaptive decision making. Thus, we adopt this approach in our inference strategy. To implement the closed-loop approach, the latest environment state must be incorporated into prompt, ensuring the L.M remains context-aware. Specifically, we define the prompt text as follows.

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Dynamic states streaming Payn: At time step t, the prompt context P dyn = {(veh, pass,), where veh, E @l denotes vehicle state vector, such as positions and arrival time. pass, E 1 denotes demand tensor, including origin-destination pairs and request time.
3.2.3 Evolutionary prompt optimization
To facilitate reasoning in the LLM through experiential learning, we iteratively optimize a population of prompts using an evolutionary algorithm (EA). Departing from traditional EAs where individuals are parameter vectors, cach prompt here is treated as an evolving entity, with fitness evaluated by simulation-baset inference scores. Prompt evolution is guided by heuristics generated via the harmony search (HS) algorithm, method well-established in classical optimization.
Harmony search algorithm HS emulates the process of musical performance, aiming to achieve an optimal state of harmony. As depicted in Algorithm [ its evolution is gov- erned by two key parameters: harmony mem- ory considering rate (HMCR) and pitch adjust- ment rate (PAR). HMCR determines whether the next iteration involves an exploration branch (introducing new solutions) or an exploitation branch (refining existing solutions). exploita- tion is selected, PAR facilitates local adjust- ments with certain probability, allowing fine- tuned modifications. The prompt evolution pro- cess formalized through a population ma- trix P E IR I xT where I, N, denote itera- tion times, population size and simulation steps.
Update Population
HUCH
WAY
Randbm interonce
Heuristic Improvement
Imuuraaive Geramatiin
(Scanario Bloca)
(Sconario Btock)
(Scenerio Block:
+ (Dynamic Block)
+ Dynamic Blook
+ Civen the above Information:
+
Givnn the above informeiont
Gven the above information
+ (pperatori Blokk]
[Operater2 Block)
[Operater Brocko
(Format
(Format Biock)
Exploration
Exploitation
Figure 3: Evolutionary mechanism of harmony search algorithm
P. [
(())) (()
k = [x((0
(1) fx here X ( denotes the k-th individual at iteration i,
[x (1). ..., o (T))- The fitness landscape is defined as: f(X)= (x)=EE,. -EEDD[ (R(ED(((((X7))))) r)]], where & represents the simulation environment and R the fitness cost over trajectory т. We adapt the HS algorithm through a probabilistic transition matrix:
(()) is its fitness value, also, x
Exploration if > HMCR
II
Exploitation if S HMCR
where UO0, 1). The exploitation phase incorporates dual refinement strategies: W XW heuristic + (1 W. with À controlled by PAR.
Wesploit
As illustrated in Figure 3 varying the value of a and X results in the selection of one of three distinct prompt compositions. Each composition consists of four components: a scenario block Paees dynamic constraint block Payn. operator block defined in Table and format restriction block Prestriction with further implementation details provided in Appendix A.9.3
Table I:
Three operators for LLM-driven objective generation. Equations and example prompts illustrate the mechanisms of each operator.
Random Inference (W1)
Heuristic Improvement (W₂)
Innovative Generation (Wa)
If exploration is chosen, the LLM generates objective without historical knowledge.
The LLM generates enhanced prompt by refining heuristics derived from the parent prompt.
The LLM formulates novel prompt, drawing inspiration from the parent prompt. X LLM(Xpana)) punem Pwe, In)
X LLM(U P.w)
X LLM( purent, Pwe, IT)
Example prompt:
Example prompt:
Example prompt:
"Please generate objective for first-level assi assignment model."
"Develop improved objective function by [instruction block1]."
"Reinvent the objective function from the
previous run by
[instruction block2]. "

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Our three evolutionary operators are listed in T The instruction blocks "[instruction block and "finstruction block2]" contain specific directives assigned their respective operators, generated
by DeepSeek-R1 37 38
A detailed explanation is provided in Appendix A.9.2
Multiple queries are issued different time steps throughout a complete simulation run, thus, the operator categories at different time steps are selected collectively in Al- gorithm [ At this stage, only the parent prompt fitnessss; and operator cate- gory are chosen, while additional compo- nents, such as the scenario block, dynamic block, and format block, are incorporated to construct complete prompt when the simulation step executed. A detailed full loop algorithm of this process is provided in Appendix A.4 Algorithm [ function INDIVIDUALSELECO is trying to select an elite parent (individual with good fitness) from the old population. Each individual consists of three components: dynamic en- vironment imputs spanning T time steps; corresponding LLM response objectives, which also span 7 time steps; and the final fitness value evaluated over horizon T. As the horizon T increases, the context length for each parent becomes substantially larger. To mitigate the growing token size, the function TOKENSELECT is employed to retain only the most critical components, specifically the objective part, which is then combined with the fitness value to construct the final parent prompt.
Algorithm 1 Generate New Individual
Require: HMCR, PAR, Simulation steps
Population size N. Population Po (iK.eiissss)
(Xx, filnesss)))
Ensure: NewInd, for L
I: for step = to do if randl() > HMCR then
3:
NewInd, + {W1)
4: else
5:
(X,, fitness;) <- INDIVIDUALSBLEC((P)
6:
X; 4- (
7:
A A BernoulliPAAR)
8: NewInd, +- -W, Si)
(1 A),,
end if
10: end for
4 Experiments
4.1 Experiment Settings
Dataset and baselines The 9 testing scenarios in Table are constructed using the New York taxi dataset [39]. We also test on Chicago taxi dataset [40], experiments and details are provided Table 3 at Appendix 4.2 A comprehensive description of the scenario generation process is presented in Appendix A.3 The details of all baseline methods are provided in Appendix A.2
Evaluation metrics In the simulator, the waiting time for each passenger is computed, and the mean waiting time is used as the evaluation metric, as presented in Tables 2 and Accordingly, methods employing two-level optimization frameworks should consider passenger waiting time an objective in the second-level sequencing problem. Similarly, in the RL approach, minimizing the total time serves the reward signal for model training. For LLM-based methods, the prompt is provided to instruct the LLM to minimize passenger waiting time.
Implementation details In our experimental setup, we utilize the DeepScek-RI-Distil---eeee 41 model through the Hugging Face platform API as the default large language model for all LLM- based methods, which allow us evaluate the adaptability highlighting its potential applications. The temperature parameter is configured to 0.9. methods all executed 3 times for each scenario, and the mean value of these three runs reported in Tables and FunSearch is performed under 20 iterations. EOH and our method all employ 10 iterations with population size of 5. All optimizer- based methods, either manual objectives or our adaptive-objective method, optimization solver Gurobi [2 is adopted to solve the problem running on a PC with 13th Gen Intel Core 19- 13900KF x 32 CPU up to 5.80 GHz and RAM 32GB.
our method on smaller LLMs, thereby
2
4.2 Experiment Comparison
Baseline comparisons on Manhattan taxi dataset Table presents the experimental results on Downtown Manhattan. Our method outperforms the best baseline (FunSearch) out of scenarios, excluding small-scale cases and 3, and achieves an average improvement exceeding 14% for all

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scenarios, In scale scenarios (8 and 9), it further reduces mean passenger delay by over 40% compared to the strongest baseline (Full RL). Three key findings can be found:
Manual objective limitations: While composite objective functions that integrate distance, temporal, and utilization signals (Distance x Tempprrl x Utilization) outperform single- objective variants (c.g., 5.77 min vs. 14.88 min in Scenario 3), their effectiveness deteriorates under increased problem scale: Waiting times increase to 9.27 min in Scenario 9 (P200_- C120_0, indicating limited scalability.
RL-based and LLM-only methods:
RL performs well in low-complexity settings (1.27 min and 3.10 10 min in Scenario 1) but struggles with reward sparsity and exploration inefficiencies in larger scenarios, especially the first RL. method (10.35 min in Scenario 9). While LLM- based EoH struggles with dynamic adaptation (10.27 min in Scenario 9), Funsearch's heuristics show better generalization (8.98 min). However, both approaches lack dynamic, per-time-interval feedback and low-level optimizer to enforce solution quality, resulting in suboptimal performance compared to our hybrid method in large cases.
Hybrid LLM+opümizer approach: By coupling LLM adaptability with optimizer precision, our hybrid framework delivers consistent performance gains, particularly under high-demand, long-horizan horizon conditions (4.01 min in Scenario 9), achieving approximately 40% improve- ment over Full RL (6.82 min). The performance gap widens with scale, highlighting the importance of dynamic objective formulation and closed-loop L.LM-optimizer interaction.
Our approach succeeds by combining L.LM-driven semantic reasoning with solver-enforced mathemat- ical rigor. LLM iteratively refines high-level objectives through prompt-based updates, overcoming the rigidity of static objectives in traditional systems that lack low-level constraint awareness. Mcan- while, solvers ensure spatiotemporal feasibility and numerical precision at scale without relying on large training datasets, as required in RL. This closed-loop mechanism, adapting objectives via the LLM and enforcing feasibility via the solver, effectively balances exploration and tractability. achieving state-of- the-art performance on mobility- on-demand benchmarks. Further analysis and experiments Chicago dataset are provided in Appendix A.I.I
Table 2: Average passenger waiting time (minutes) across optimization methods and scenarios on New York taxi dataset
Category
Method
Scemario
2
4
7
Distance 8
6.31
14.78
14.38
11.42
24.67
22.32
18.08
$4.98
28.04
Distance Utliization
5,02
7.38
6.96
9.07
9.89
7.05
11.59
Mansal Objectives
Tampoeal Uniiization
10.09
8.81
9:93
9.74
11.48
15.23
10:97
13.26
18.99
Distance
Temporal Utiization
3,32
4.37
5.19
9.27
RF. Methock
Full RL
Defullt
X
Seq"
3,10
1.27
2.35
3.98
2.42
4.92
3.50
4.83
3.20
7.48
4.78
10.35
6,82
LLM Methods
FunSearch
13
0.77
1.74
1.55
5.29
2.95
5.43
5.78
5:79
10.27
1.64
3.04
3.87
8.98
Hybrid (IIM-Optimizee)
Ours
1.55
1.37
2.50
1.89
2.59
4.14
3.10
2.25
4.01
Distance: Travel time between vehicle start position and passenger pickup&atropof location.
Temporal: Gap between the vehicle non-idle time and passenger request time.
Utilization: Taxi service efficiency (vchiclesrrquuss))
Default: Use default objective (Distance x Temperal x Utlization) in first-level ussignment optimizationn RL-Sey Reinfoncement learning method udopted to solve second level sequencing problem.
Baseline comparisons on Chicago taxi dataset Due to the relatively larger spatial extent of Chicago's zones compared to Manhattan, combined with the highly imbalanced distribution of ride requests (Appendix A.3 for details), the average digtinddstination (OD) travel times in Chicago are significantly longer. This increased travel distance contributes to overall higher passenger waiting times across the scenarios in Table Table demonstrates our framework's 's consistent superiority on Chicago distribution conditions, reducing mean passenger waiting times by an average of 18% in large-scale scenarios (e.g., 10.79 min vs. 14.35 min Scenario 7, 14.51 min vs. 15.93 min in Scenario 8, 15.37 min vs. 19.79 min in Scenario 9). Compared to the best baseline (FunSearch), our method outperforms in 8 of 9 scenarios, with a marginal exception in small-scale case 3, and delivers an average improvement exceeding 18% across all cases. Three critical insights can be found:
8

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Manual objective limitations: Composite objectives (Distance x Temporal x Utilization) degrade severely under scale, with delays escalating 25.01 min in Scenario 9 (vs. 15.37 min for our proposed approach). When the trip distribution is significantly imbalanced, as shown in Appendix AA.3 single-objective variants that optimize solely for travel distance perform poorly under high-demand conditions, resulting in extreme delays (145.14 min in Scenario 8).
RL-based and LLM-only methods: Full RL outperforms Default+RL-Seq (20.03 min vs. 24.63 min) but remains inferior to our proposed methods, with a gap of 4.66 min compared to the best-performing approach (15.37 min). This underscores RL's sensitivity to reward design and training data coverage. LLM-only methods exhibit inconsistent adaptation. Specifically, FunSearch outperforms EoH smaller-scale scenarios (Scenarios 1-3), whereas EoH achieves better results in larger-scale settings (Scenarios 7-9). However, both methods lack dynamic feedback mechanisms and a fine-grained optimization layer ensure solution quality, leading to inferior performance relative to our proposed hybrid approach.
Hybrid I.LM+optimizer approach: Our framework delivers robust performance across all scenarios, with several cases exhibiting substantial gains. For instance, in Scenario 4, our method achieves an average delay of 8.40 min compared to 12.05 min with the Full RL; in Scenario 7, 10.79 min vs. 14.35 min with the Full RL; and in Scenario 9, 15.37 min compared to 19.79 min with EoH. These improvements, all exceeding 20%, underscore the importance of closed-loop adaptation in dynamic and high-demand environments.
These results generalize the findings from the Manhattan dataset (Table 2) proving our method's adaptability to varying urban layouts and demand distributions. The advantage large-scale scenarios underscores the necessity of iterative LLM-optimizer interaction for real- worldw ride-hailing systems.
Table 3: Average passenger waiting time (minutes) across optimization methods and scenarios on Chicago taxi dataset
Calegory
Method
Secnriio
Distance
35.87
47.85
41.08
121.41
145.14
100.44
Distance x Udlizatioo"
11:60
15:34
18:35
15:70
18.50
22.77
19.07
22.74
90.38
Manual Objectives
Temgonal Diliicaion"
16.90
17.33
23.96
Distanes x Toopocal x Utiization
9.54
13.79
18.19
11.81
16.71
20.74
17.21
20.68
25.01
RL. Methds
Defini RE-S
10:6
10.87
15:37
19:27
18.68
21:57
24.00
34.86
24.63
Full RL
10.33
13:43
15:40
12.05
14.43
16.50
14.35
15.93
20,13
LLM Metbods
FenStorch H
9.03
10.70
16.87
21.70
12.82
15:35
18.29
17.42
21.24
FaH
13.05
17.16
Reld (LLM) (LLMM-ppliminn Oplimizer)
Oura
8.65
9.58
11.30
8.40
12,32
14.96
10.79
14.51
15.37
Distance:
Travel time between vehicle start position and passenger pickup&ddropoff location.
Temporal:
Gap between vehicle non idle time and passenger request time.
Utlizaaion: Taxi service efficiency vehhiclesrequeee).
Default: use default objective (Distance x Temporal x Utilization) in first-level assignment optimization. RL-Seq: reinforeement learning methed adopted solve second-level sequencing problem.
4.3 Ablation Study
Scenario prompt composition In- corporating Pmodei which encodes structural blueprint of the optimization model, significantly reduces waiting times (e.g., 2.86+10.29 min for 770_C60_T6600 though variability increases due to
Table 4: Average waiting time (minutes) by prompt composition
Prompt Composition P50, 130, T300
170,
_60__66___0 T600
P100_C80_T0 CS0_CCC_T_00 T900
P130, (80 T1200 C22000000000000
Pas U Pao U Pau
9.93 + 0.00
5.86 + 0.00
5.29 + 0.00
6.46 + 0.00
Pa, U Poo Panna
8.72 - 0.91
2.86 = 0.29
3.07 0.89
4.54 0.36
+P +Paancmn
8.16 + 1.48
4.12 0.83
2.59 0.52
2.25 + 0.05
Note: Mean = standard deviation across runs Bold: best per scenario
(P=Passengers, C=Taxis, T=Time(s)). Prompt variants: P U P gro U Pus:: System+Geo+Data structure (Sec. 3.2.1 Pas U 200 Pmodel:
System+Gro+MMddel structure (Sec. 3.2.1
PPessniction denotes Pays U
Pos U model U Preaaittion: System+Geo+Modee Moweeee-ooeel--eiiii-i-ooooo- Gulo-iiupaiicceiiblleeee constraints
(App. A.9.3). Paua defines variable formats, Pmods specifies full MILP struc- ture.

---

LLM- -LL--gnnrrted function diversity. Further integrating P Prestriction (Gurobi curoob-coompttble constraints) achieves the lowest costs three compositions (e.g,, 2.25=0.05 min for P130_C80 (C0_TT2200)
T compares the performance of three prompt compositions involving scenario block and format block, with means and standard deviations computed over three runs (visualized in Figure When only basic system parameters and data specifications are provided, the model produces high passenger waiting times (e.g.. 9.93.10.00 min for P50_ _30 _30__T30) with zero standard deviation across runs. This consistency is attributed to the generation of invalid objective functions, which are rejected during Gurobi's 's model verification; as a fallback, a static default objective applied (see Table 6 in the Appendix).
C
ClaR
(a) Run
(b) Run
(c) Run
Figure 4: Average passenger waiting time (minutes) under different compositions and runs. Cases: [P50_C30 (300), B (P70_C60 (600), C (P100_C80_TSOO and D (P130 _ap___o_____ T1200). Blue line
(Psys Ugoo Pana), orange line (Psys U Pgeo Pmodel), green line (Pays P Pgeo U Pana U restriction))
Figure 4 illlutrrates the average passenger waiting time corresponding Table When problem scale is small (Case A and Case B), Pmodel is enough, but as complexity increases, P Pestriction becomes essential. This aligns with Table 6 where larger scenarios exhibit higher error rates for compositions lacking constraints (e.g., 36.7% errors in P130 _ without P Prestrictioa) necessitating explicit formalization to stabilize LLM outputs. This underscores the necessity structured task grounding (model definitions) and constraint formalization for reliable LLM-driven optimization.
Dynamic environment feedback impacts shown in T we perform experiments on two query mechanisms - single time on first time step and multiple times on each time step at scenario
Table 5: Average passenger waiting time comparison: open-loop vs. dynamic closed-loop query mechanisms
Query Mechanism
Runf
Run2
Run3
Avg.
_ 2100, the final result of each run at last iteration is provided. Clearly, the dynamic multi-time time query demonstrates superior performance compared to the dingle-tmme query. The dynamic approach benefits from frequent interactions with the LLM, allowing it acquire rich semantic information each step. In contrast, the single query only provides the LLM with the initial environment setup, limiting its ability to propose an objective based evolving information. This results in the generation of identical final outcomes, highlighting the limited flexibility of the single-query approach. Further discussion of cost reduction provided in Figure 9 in Appendix A.I.I
Single open-loop
4.62
4.62
4.62
4.62
Dynamic closed-loop
2.23
2.31
2.19
2.24
Note: Values minutes. Bold entries highlight the
superior performance of the dynamic closed-loop mech- anism.
Conclusion
In this paper, to solve the dynamic dispatching problem in ride-hailing system, we propose hybrid approach that combines LLM and optimizer in a dynamic hierarchical system, where LLM-generated objective in the high-level model is served as guiding heuristics for the low-level routing optimizer. High-level objectives are iteratively refined in closed-loop evolutionary process, with performance evaluated across simulation epochs. A harmony search algorithm adaptively explores the LLM prompt space to improve objective quality over time. Experiments on Manhattan downtown and Central Chicago taxi datasets demonstrate the effectiveness of our approach, achieving an average 16% improvement over state-of -rrt---rttt-tttt baselines. Additional results, future work, and supporting materials are provided in Appendix A The source code can be found in: https: sattps://ge htt///ttttt tiut..u..... com/ yi vizhangele// 1mm guided mod- opt.mizatiinnnnn
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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The abstract and introduction accurately reflect the paper's core contributions and scope. They clearly state the main methodological innovation: hybrid framework that integrates LLM with mathematical optimization for dynamic recision-makingg in ide-hailing platforms. The abstract and introduction appropriately highlights the limitations of existing RL and decomposed optimization methods. They also outlines key components such as the training-free nature of the method, prompt-based high-level objective generation, and the closed-loop refinement mechanism via harmony search. These contributions are thoroughly developed in the main paper (Section 3.1 and Section [3.2] and Appendix Algorithm A.44 and evaluated through experiments (Section 4 Appendix IA. on real-world datasets (New York and Chicago taxi data). The claims of performance improvements and remantic-guided optimization are substantiated by the results, making the abstract and introduction faithful summary of the paper's contributions.
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Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: Appendix A.1.2 discusses several limitations and outlines directions for future work, reflecting a clear understanding the boundaries of the current study. Specifically, Appendix 1.2 acknowledge the limitation of simulation fidelity, noting that the current experiments do not incorporate fine-grained, real-world traffic dynamics such as congestion patterns, signal timing, or road disruptions. To address this, propose integrating micro- scopic traffic simulators like SUMO to enable gigher-fidelity evaluations under realistic operational conditions.
Moreover, we recognize the static role of the LLM in the current framework: it is not fine-tuned and adapts only through prompt-level feedback. Thus, we propose future en- hancements involving reinforcement learning techniques enable co-adaptation between the LLM and the optimizer, treating the optimizer as critic to fine-tune the LLM more efficiently.
Finally, we identify the potential for quantum computing to overcome current limitations in scalability and solution efficiency. We envision integrating guantum-enhancee solvers into the framework to accelerate both data-driven reasoning (via LLM) and combinatorial optimization, allowing tighter interaction and real-time deployment.
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should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, c.g., if the approach was only tested few datasets with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
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3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
The paper includes formal mixed-integer logic model formulation of the ride- hailing problem, which is introduced and discussed in the Appendix A.6 The conversion of logical constraints into equivalent mixed-integer linear constraints is also detailed, with accompanying proofs provided
Justification:
Appendix AA. to ensure correctness. While the main paper focuses on the hierarchical decomposition into high-level assignment and low-level routing problems, as part of the I.L. L.g--dddeed optimization loop, clearly states the modeling assumptions and system-level structure. Additionally, the theoretical foundations behind the modular decomposition and prompt-base refinement via harmony search are supported by formal modeling and complete derivations in the Appendix.
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Question:
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Section 4.J presents the experimental setup, evaluation metrics, and imple- mentation details. The New York and Chicago taxi datasets used in the experiments are described in Appendix A.3 Benchmark comparisons with baseline methods are provided in Table and Figure 5 for the New York dataset (main paper), and in Table B and Figure 6 for
Justification:
the Chicago dataset.
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to support fu- Ablation studies are conducted to evaluate several key aspects of the system. The impact of scenario prompt composition is analyzed in Table and Figure _ The effect of dynamic environment feedback mechanisms is detailed in Table 5 Additional analyses presented in the appendix include: cost variations over iterations for both multi-query and single- query settings (FigureP)) an evolutionary hyperparameter study (Figures TO and TT); an investigation of LLM temperature sensitivity (Figure 12); and an analysis of LLM-innuued error rates (Ta
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Justification: While our work does not involve training machine learning models in the traditional sense, it integrates a well-established pretrained LLM with a mathematical op- timization framework. As such, there are no training/test data splits or model training procedures to report. However, to ensure clarity of results, we provide: (1) Full implementa- tion details for all components involved in our approach in Section 4.1 Implementation
details; (2) The New York and Chicago taxi datasets, along with the methodology for
generating the nine testing scenarios, are described Appendix A3.3 (3) To evolve prompt
heuristics for the LLM, we utilize the harmony search algorithm. Key hyperparameters of this algorithm (e.g.. HMCR and PAR) are analyzed via sensitivity study presented in the Figures 10 and 11 at Appendix A.I.I These details collectively provide complete picture
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(1) We benchmark our method on two distinct real-world taxi datasets (New York and Chicago), providing evidence of adaptability to divergent trip distributions.
(2) We evaluate the impact of prompt composition LLM inference stability. Specif- ically, Table 4 reports average passenger waiting times across four scenarios and three prompt variants, showing mean and standard deviation over 3 independent runs per setting. These error bars reflect variation in LLM outputs due to prompt randomness and inherent nondeterminism.
(3) We study LLM output validity through an error rate analysis (Table 6 in Appendix A.I.I. showing acceptance rates of L.LM-generaed objectives by the Gurobi solver across prompt types and scenarios. Each configuration evaluated over three independent runs, and average error rates are reported.
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Default RL-Seq: This method employs deep reinforcement learning for low-level sequencing. We have cited the relevant publication in our manuscript and utilized a publicly available implementation from GitHub. Although the repository does not explicitly specify a license, we have reached out to the repository's 's author to clarify the terms of use.
tree/master
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https://github. com/ cal www.pringgg///meemmmmm..m.... MDDg//DDD////// FPOOD/t2 tttteeeeemnnnee
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https://github. Foom/FeiLiuse http:///H//////eeeeeeeeeeeeeeeee www.d//maain/iiaea CENNSE
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https://gitthe httpab/.....eeelaaat////////lla/
L LICENSE
This dataset is publicly available and maintained by the NYC Taxi and Limousine Commission. It is accessible via the NYC open data portal and is provided under the NYC Open Data Terms of Use, which generally allow use, modification, and redistribution for both commercial and non-commereial purposes, with attribution.
New York Taxi Dataset:
https: wwt.:ccewyyoooooooooooo us/ www.ovvwwwwwwwwweevoooooooo
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Taxi Taae:-Trrpp-2203332022///eereerr
disttaimss html
We utilize this publicly available large language model via the Hugging Face platform API for prompt optimization and L.L.-base decision making. The model is released under the MIT License. DeepSeek- - D-iitilll-- Qwen-32B derived from the Qwen-2.5 series (originally licensed under the Apache 2.0 License) and further fine-tuned on 800k curated samples by DeepSeek-R1. We cite the official model repository in our manuscript and adhere to all licensing terms. https://hugginas coleeeeeco//eeeeeeeeeeeeeeeeeeee deepsee ai/DeepSee- R1 Distilllllll- -oa---een-322
RrepSeek-Ri--iistll-----eeee
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A Technical Appendices
A.1 Discussion
A.1.1 Further studies
Spatiotemporal analysis To il- lustrate the spatial-temporal dis- tribution of passenger waiting times across different methods, Figure 7 and Figure 8 present the results for Scenario 9 (as de- fined in Table and Table 3) for Downtown Manhattan and Cen- tral Chicago, respectively, where 200 passenger requests are dis- patched within a 1200-second time window and served by 100 taxis. To facilitate spatial inter- pretation of the study area, Fig- and shoow the zone-based maps of Downtown Manhattan and extended Central Chicago, respectively, with each study area partitioned into 19 predefined zones used as both origins and destinations for passenger trips.
Zones
Alphobet Cityiznn 1)
Battery Parkipone 12)
Park Cityzona
Vilapelzone
Rnandal Distritt North 87)
Finandsl Dieerve So Southppone R8)
Ludeon Pagzone
Groons Village South(eone 114)
Little Itatyhoounstzons 1+1)
Lower Tast Sidezoone 148)
Nautpadonn// Village WaatCoone 138)
New York
Seaporttzone
SoHaCZone 213)
Marryttan Bridac)
TrBeCa/C Conter(rone 2313
The fnddiee/SSwwer Park(cone 232)
West Villageçcone 249)
World Center(zone
Figure 5: Hone-based Manhattan downtown map
Compared to the zoning structure of Manhattan, Chicago's 's zones, also referred to community areas, are generally larger in spatial extent, es- pecially outside the central business district. In particular, OD trip patterns reveal a notable con- trast between the two cities: while Manhattan demonstrates relatively balanced distribution of taxi trips across its zones, Chicago's taxi ac- tivity is highly concentrated within a few central areas, most prominently in zones 32, and 28. Further details on the OD distribution are pro- vided in Appendix A..
DOUGLAS(zone 35)
GAKLAND(zane
BOULEVARD 30)
KBNW000022re 39)
WASHINGTON
IIDE PARK(zOne 41]
LOGAN SQUARE(2one 22)
TowNN
NEAR WCST SiDt/sone 20)
LOWER WEST SiDE 31)
SOUTH SiDE
ARMOUR SqueRR 34)
NEAR NORTH SIDE 0)
KOCKLEEY pARK) 59)
LAKE VIEwi/zone
Figures and illustrate the spatiotemporal distribution of passenger waiting delays across zones in Downtown Manhattan and Central Chicago, respectively. Both figures correspond to the high-demand Scenario 9, as defined in Table 2 and Table The passenger waiting de- lay is computed as delay, = mazx(fpick tre9, .). where ppick denotes the vehicle pickup time and eye9 the passenger request time. Delays are aggregated into 600-second temporal bins (e.g..
CENTER(cone
BAIDGEPORTT/e 60)
LINCOUN PARK(zone 7)
Figure 6: Zone-based Chicago extended central map
p E [600s, 1200s) maps to Slot 1) and 800-second temporal bins (e.g., 1 spick E [800s, 1600s] maps to Slot 1) for Figures and respectively. Based on the aggregated bins, delays are spatially averaged over a set of predefined geographic zones. This joint spatiotemporal aggregation captures localized service inefficiencies while preserving the dynamics of demand-supply imbalance over time. The heatmaps quantitatively validate our framework's 's advantage: under identical input conditions, our method completes all passenger pickups within 4 time slots (4x 600 seconds for Figure 7 4x800 seconds for Figure 8). whereas baseline methods require 5 to 7 time slots in the Manhattan setting and 5 to time slots in the Chicago setting fulfill all requests. As time progresses, the delays of unserved passengers accumulate, resulting in significantly higher average waiting times in the later time slots. This effect is reflected in the heatmap as progressively intensified red shading in the final temporal bins. As manual objective methods (Tables and often yield high passenger delays, only their perforrefnn approach (Distance x Temporal Utilization) is included in Figures and for comparison.
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NAC
NAC
MAC
WEA
NAT CIPED DHENU
OHIC FOODA
(TICK -OOES
The
ISTED SIERA
CIRSS CHODL
tee
SACT
Figure 7: Spatial-temporal distribution of passenger waiting times across optimization methods in Scenario (P200_ L1100_C110__111 _ under New York taxi dataset. Each row represents method: (1) Manual composite objective (Distance x Temporal Utilization), (2) Default RL-Seq. (3) Full RL, (4) FunSearch, (5) EoH, (6) Our LLM A.t-Oppiiizerr approach. Columns show 600-second time windows. Heatmap colors show average waiting times per zone.
In Figure first approach (Row 1) exhibit persistent hotspots for Zones 144, 79 and 249, due to static designed objective. RL methods (row 2-3) show temporal degradation, especially row 2. where expanding high-delay regions in Slot (3000-3600s) highlight RL's limitations in enforcing hard constraints (e.g., detour limits) and generalize to unseen demand patterns. LLM-bbsed methods, notably EoH, perform moderately better, completing all requests within time slots. Although FunSearch is the best overall baseline across scenarios, as it outperforms our method in Scenario I (0.77 min) and Scenario 3 (1.55 min) as shown Table [ its performance degrades significantly under high-demand conditions, particularly in Scenario 9, captured in Figure 7 In contrast, our proposed framework completes all requests within 4 time slots while maintaining spatially consistent low delays (light yellows) through closed-loop LLMMoopimizzrr interaction. This analysis quantifies how our framework overcomes RL's data dependency, the rigidity manually decomposed objectives, and the constraint unawareness of LLM-only methods. The heatmaps align with Scenario 9 results Table 2 proving that semantic reasoning (LLMs) and symbolic grounding (solvers) effectively mitigate delay accumulation in large-scale urban mobility systems.
similar pattern is observed in Figure 8 Unlike the results on the Manhattan dataset, the Chicago scenario requires more time to complete all requests, primarily due to more imbalanced trip distribution and longer average travel times between origin-destination pairs. Specifically, the first (Row 1) and second (Row 2) baseline approaches require 9 and 8 time slots to serve all passengers, respectively. Full RL (Row 3) and EoH (Row 5) demonstrate improved performance, completing the task in 5 time slots, while FunSearch (Row 4) shows moderate degradation with 6 time slots. In contrast, proposed method completes all requests within only time slots, indicating enhanced adaptability to complex urban topologies and demand heterogeneity. This highlights the effectiveness of integrating I.LM as a data-driven prior with mathematical optimization for robust and efficient decision-makee making..

---

SMIS
C9KE
2et
PAL
Figure 8: Spatial-temporal distribution of passenger waiting times across optimization methods in Scenario 9 (P200_co __1111000__0001 C100_T1200) under Chicago taxi dataset. Each row represents a method: (1) Manual composite objective (Distance x Temporal x Utilization), (2) Default + RL-Seq, (3) Full RL, (4) FunSearch, (5) EoH, (6) Our LLM-Optimizer approach. Columns show 800-second time windows. Heatmap colors show average waiting times per zone.
Query comparison Figure D illustrates the cost changes under different iterations for multi-query and single-query, corresponding the scenarios described in Table Across all runs, multi-query achieves steeper convergence trajectories, reducing costs by an average of 50% (relative to single- query) within 10 iterations, which aligns with Table This results from the closed-loop interaction between the LLM and the optimizer in the multi-query framework, in contrast to the open-loop configuration of the single-query approach.
Average Waiting Time (min)
Ausrage Weiting Time (min)
Average Waiding Time (min))
4.5
4.5
4.0
4.0
4.0
3.5
Single Call_ Cnil_Run]
Single Rall
3.5
Single Call, Cail_Run) Run3
3.0
3.0
2.5
2.5
2.5
10
10
Itaration Numbar
Iteration Number
Iteration Number
(a) Runl
(b) Run2
(c) Run3
Figure 9; Cost changes through iterations between multi-call and single-call approaches.
Evolutionary hyperparameter study As captured in Figure 10 a grid search is performed for the hyperparameters HMCR and PAR in the harmony search algorithm for P100 C80 _ The HMCR values tested are 0. 0.5, and 0.9, while PAR values tested are 0.2, 0.5, 0.8. Figure [0a and Figure depict the best individual cost in last iteration and the mean cost of entire population last iteration. The results demonstrate significant performance improvement under high HMCR (HMCR 0.9), suggesting effective exploitation of high-quality parent solutions from memory. This
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3.34
2.36
2.42
3.04
2.86
3:34
2:7g
2:46
0.3
3.14
3:43
3:33
2.62
2.70
0.5 HMCR
0.5 HMCR
(a) Min cost at last iteration
(b) Mean cost at last iteration
Figure 10: Hyperparameter sensitivity analysis on HMCR and PAR Cost at last iteration.
aligns with our hypothesis that preserving promising solution components through frequent memory recall (via operator W2 and W3) substantially enhances convergence properties. On the other hand, while lower PAR value (0.2) yield optimal results at HMCR 0.9, this relationship no longer exists for HMCR < 0.5. Consequently, operator selection frequency between W2 and W3 appears governed more by solution quality feedback than fixed parameterization. A similar pattern in cost reduction with significant decrease in cost when HMCR is high.
observed in Figure Ila
[lb and
11c.
Avarage Waiting Time (min)
Average Waiting Time (min)
Average Waiting Time (min)
0.8
HPCC Levels
HMCR--0.1
IMMCR-0.5
1MMCR=0.9
HMCH Levch
HMCR Leach
HMCR=0011
HNCR=0.1
3.0
HMCR+0.9
HMCR-0.5
HNCR-0.5
HNCR-0.9
2.8
2.8
10
10
5
10
Iteraioo Number
Iteration Number
Iteration Number
(a) PAR is 0.2
(b) PAR is 0.5
(c) PAR 0.8
Figure 11: Hyperparameter sensitivity analysis on HMCR and PAR Costs at various iterations.
LLM temperature study We evaluate the impact LLM temperature (r {0.1, 0.5, 0.9, 1.3)) on optimization efficiency across three scenarios, captured in Figure T2 The contour plots depict cost landscapes over 10 iterations and 4 temperatures, revealing that extreme temperatures 1.3) consistently lead to higher overall costs, whereas = 0.9 yields the
(T = 0.1 and lowest values, indicating more effective optimization. Across all settings, convergence occurs rapidly, with trajectories stabilizing within 2-3 iterations. When 1.3. high stochasticity disrupts objective function coherence, generating infeasible or unstable formulations. While r = 0.1 overly constrains diversity, causing premature convergence suboptimal basins. The intermediate setting of r = 0.9 strikes an effective balance between exploitation (leveraging high- high-likelihood tokens for constraint adherence) and exploration (sampling novel objective structures). These findings highlight temperature as a critical hyperparameter for regulating I.LMMoptimzzer symbiosis.
Error rate study As discussed in experiment 4.3 Table 6 quantifies the failure rates of LLM- generated objective functions across four scenarios and three prompt compositions, measured at the last iteration (10th iteration). For each scenario, LLM queries scale with time window duration: P50 C30 T300 (1 query per individual, 5 total responses for entire popsize) to P130 C80 T1200 (4 queries per individual, 20 total responses). Error rates are computed as invalid objectives
responses
When only Psy. U P are provided, error rates reach 100% in all scenarios, as inadequate prompts yield semantically invalid formulations, Replacing Paata with Piootel reduces errors 20% (P50) and 7.7% (P130), demonstrating the necessity of task grounding via structural priors. Further
26

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(a) P50_C30_T3M0
(b) P70_C60 CC0__66000
(c) 1100_C80_9900
Figure 12: Cost landscape across temperature settings and iterations.
incorporation Prestriction minimizes errors to 0.0% in smaller scenarios (P50-P100) and 8.3% in P130, highlighting the role of constraint formalization. This indicates the critical role of progressive prompt enrichment, via model structuring and constraint formalization, in ensuring feasible and valid outputs from LLM-driven optimization pipelines. Additionally, for the latter two prompt compositions, error rates exhibit an upward trend with increasing scenario complexity (rising from 20.0% in P50 to 36.7% in P130 for Pmodel 0.0% in P50 to 8.3% in P130 for P Prestriction); reflecting the escalating fragility of LLM --M--eeerated objectives in high-dimensional spaces.
Table 6: Error rate analysis of L.M--genrraed solutions across prompt compositions
Scenario
Prompt Composition
Runl
Run2
Run3
Avg.
as U g00
Pama
100%
100%
100%
100.0%
PS0 C30_ T300
Pus U goo Pmndel
0%
60%
20.0%
Premncoon
0%
0%
0%
0.0%
Pas U Paim
90%
100%
100%
96.7%
P70 170_C60_T600
P P U
Pmudd
0%
0%
0%
0.0%
+ P Prestincon
0%
0%
0%
0.0%
P U U Pasna
100%
100%
100%
100.0%
P100_C80_T90N
Pas U U Pmndel
13.3%
60%
26. 7%
33.3%
P Peanitionn
0%
0%
0%
0.0%
P UP geo U Patta P U Pmnddl
100%
100%
100%
100.0%
1180_C8O_T1200
55%
25%
30%
36.7%
Presriation
15%
10%
8.3%
Note:
Scenarios denote passenger counts (P), taxis (C), and time windows (T). Prompt components: system
(Poy),, geometry (P (.,,), argument/variable format(Pana. model blueprint (P (Pmodet), and constraints (P (Prestiction). +P Pressticion denotes Pay. U Ppo U ' Pmmdee U P Prestriction
A.1.2 Future work
Application layer scalability and simulation fidelity Our experiments demonstrate effective optimization using 32B-parame -aramaaee LLM, suggesting smaller models can achieve comparable performance when properly constrained to domain microscopic traffic simulators (e-g.. SUMO (43) to enable high-fidelity modeling of road network dynamics, including congestion, signal timing, and stochastic travel times with spatial correlations. This integration will support evaluation of our framework under realistic operational conditions, accounting for transient disruptions (c.g., accidents, road closures) that require dynamic re-planning.
reasoning. In the future, we will integrate
Reinforcement learning for LLM-Optimizer co-adaptation The current framework treats the LLM as a fixed agent that iteratively adapts its objective-generatie behavior through prompt-base based interactions, does not modify the model's internal parameters. However, the structure of our system, where the L.MM proposes objectives and an extemnl optimizer evaluates them, mirrors the actor-critic paradigm in reinforcement learning. This opens a promising direction: leveraging optimizer feedback (c.g., fitness costs) to directly fine-tune the LLM, potentially enabling efficient training with smaller
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models. By integrating the optimizer as a critic that evaluates the LLM's proposed objectives, we can move beyond prompt engineering and into setting where the LLM is explicitly trained generate more effective objectives over time. Given previous success in using RI. to effectively fine-tune L.M.s 44. such optimizer-guides fine-tuning could enable more sample-efficient and domain-adaptive optimization pipelines, further bridging large language models and traditional solvers.
Quantum-enhanced hierarchical optimization The proposed framework leverages LLM and classical optimization solvers in hierarchical interaction to address sequential decision-making problems, we envision that advances in quantum computing could significantly enhance this paradigm. Although quantum computing still in its early stages, has shown strong potential in solving combinatorial and discrete optimization problems more efficiently than classical computers. As quantum hardware matures, it could accelerate both sides of the LLM-optimizer interaction, providing faster inference for LLM- LLM-guided heuristics and more efficient solutions from quantum-enhanced optimizers. This would enable tighter integration and more frequent interactions between the reasoning and solving components, opening the door to real-time implementations of our framework and its deployment in dynamic decision environments.
A.2 Baseline Methods
Manual-designed objectives: We construct set of handcrafted objective baselines by combining different dispatching priorities (e.g.. Distance, Temporal, and Utilization) motivated by two prior works: 2 which proposes distance-based assignment in modeling (used our Distance-only baseline), and 24 which introduces MILP models incorporating temporal and utilization aspects (reflected in our Temporal Utilization baseline). Building on these, we also include two additional combinations: Distance x Utilization, Distance x Temporal x Utilization. These rule-based objectives form the manual objectives baselines used evaluate performance across various dispatching scenarios. The second-level routing adopts the optimization model from formulation !
Default RL-Seq: As the name suggests, Default RL-Seq uses the default objective Distance x Temporal x UUtlizzation for first-level assignment. However, for second-level routing, it utilizes the RL-based method proposed in 23 instead. More specifically, we trained policy model to stochastically predict the order to pick up passengers grouped together after first-level assignment optimization, minimizing the total travel time. In our implementation, for both passengers and taxis, we use travel times to and from all possible locations as their input features to the policy model. And we use an architecture similar (23) but disable glimpsing to enhance learning. During inference, a search-based strategy is used to decide the final pickup order.
FullRL: FullRL is an implementation of an online order dispatch policy (6). where decisions are made in a rolling horizon based on advantage function estimation Ar (f. = > (st,) ) V(s:) + R, (j). where y is the discount factor, and A., is the picking up time for the matched pair of driver i and order j. The reward R., (j) is the reward for dispatching driver i to order j. To sync with the problem settings in this paper, we modify the reward to reflect the waiting time of order j instcad of using the default value of 1. Consequently, the objective shifts from maximizing the service rate (as in the original paper) to minimizing total waiting times. The current state of a driver i is represented as two-dimensional vector (location and time). The value function of driver's 's current state negatively impacts the advantage function (s:)). An action that assigns an order to a driver whose destination is in more valuable region results in higher advantages, denoted as V(S),). ). Finally, dispatching decisions at each timestamp are based on advantage function estimation.
FunSearch: FunSearch 155 utilizes LLM as well to produce methods for given problems. During testing, we made several adjustments to adapt to our laxi-passenger assignment problem. More specifically, expanding generic prompt, we provided information method input and output formats and more clearly defined the optimization purpose. Furthermore, we also adjusted the hyperparameters to encourage more explorations while searching for the final method. For each scenario, we sampled 20 methods before reporting the final results.
EOH EoH [16] is a framework that combines LLM and evolutionary computation to automate heuristic design. It utilizes five prompt-driven strategies and expresses heuristic ideas as natural language 'thoughts'. These thoughts are then converted by LLM into executable code, which helps improve and refine the original heuristics. In our experiments, we evaluate EoH on the taxi-passenger assignment problem using a constrained functional prompt. The goal is to minimize passenger waiting time while
28

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ensuring balanced and efficient taxi utilization. We conduct evaluations across 9 scenarios, with 10 evolutionary iterations and population size of 5.
A.3 Dataset Description and Testing Scenario Generation
We use the New York taxi trip data from January to June 2024, publicly available from the NYC taxi and Limousine commission (https://www .ww. .ov...oo/aae www.ggov///stieee httss/itttttttttttt//// r-c-trip-recordee data. page), and Chicago taxi trip data from January to June 2023, publicly available from Chicago data portal lhttps://datee cityof
org/ www.ooogg//rrrre. Transportation/ Taxi-Trips-2013 2013 2013-22022/w/2-----2 -etee//wVzz---eee-e tassew//bouut about data), to generate realistic simulation scenarios. For Manhattan, our analysis focuses trips that originate and terminate within the downtown area. For Chicago, we focus on central district including Loop community (zone 32) and 18 neighhoring communities. Rather than replaying historical trajectories, we extract key statistical properties from the dataset and use them to generate randomized scenarios.
Figure 13a illustrates the average spatial distribution of taxi trips the six-month period in Downtown Manhattan area. Based on this distribution, we define peak hours as the time interval from 12:00 a.m. to 10:00 p.m. We further compute the empirical frequency of each OD pair, shown in Figure 13b, For each synthetic scenario, travel tasks are sampled according to the normalized OD frequency distribution, thereby preserving realistic spatial and temporal patterns observed in the real-world data.
In Figure 14a we show similar statistics of Chicago central district. The peaks hours in these arcas are defined as from 8:00 a.m. to 7:00 p.m. Also, notice that taxi trips are highly concentrated on origin-destination pair (32, 8) and (32, 28), which is due to the fact that Loop (zone 32) is the central business district of Chicago and Near North Side (zone 8) and Near West Side (zone 28) are two regions next to Loop with large population.
(a) Manhattan downtown average task distribution
Time
Frequency Marii
Froer Cartee
Froes
(b) Manhattan downtown travel and frequency
Figure 13: Average task (passenger request) distribution and travel time and frequency statistics in Manhattan downtown
We construct two sets of nine synthetic simulation scenarios, derived separately from the statistical distributions of the New York TLC trip data and the Chicago taxi trip data. Each set consists of scenarios that share identical configurations in terms of passenger demand, taxi fleet size, and request
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Teak Sregurrcy Cemtacionn by Hour
(a) Chicago central district average task distribution
Traal Time Narts
Thees Freeweny Warb
(b) Chicago central district travel time and frequency
Figure 14: Average task (passenger request) distribution and travel time and frequency statistics in Chicago extended central district
time windows, but differ in spatial and temporal distribution characteristics specific to each city. Each scenario is parameterized by the following variables: (1) Passenger volume (P): Ranging from
35 to 200 requests, reflecting varying demand intensities. (2) Taxi fleet size (C): From 60 to 100 vehicles, testing scalability under resource constraints. (3) Request time window (T): 600s (10min), 900s (15min), or 1200s (20min), is the time window for distributing passenger request. Scenarios 1-9 in Table and Table corresponds to P35_C60 T600. 265_C80 T600, P100_C100 C000 C1000T600, _60__600, T900, $100_C80 T900, P150_ C100 T900, P70_C66 C60_T1200 P130_ T1200, P200_- C100_TI200 T1200. For example, P200 C100 T1200 denotes a simulation scenario with 200 passenger requests occurring within 20-minute time window, 100 available taxis in the system, and stochastic request generation based on the origin-destination distribution. The scenarios incrementally stress-test the system along these dimensions (Tables 2 and 3), avoiding historical replay while maintaining fidelity to real-world trip dynamics.
P50 P50_C60
A.4 Full Loop of Algorithm
The complete algorithmic process is outlined Algorithm
A.5 Search Space of Dynamic Systems
dynamic system in real-time decision makion (c.g., ride-hailing dispatching) is characterized by time-evolving states and sequential interdependent decisions. Unlike static OR problems (c.g.. traveling salesman problem or bin packing), dynamic systems require continuous adaptation to changing environments, leading to exponentially growing search spaces. The problem at time can be formulated as: S, = (Ve. C, Dr). where S, is the system state at time f, is composed by the vehicle set V/ (taxis with locations and status), the passenger set P, (pending passenger requests with pickup/dropoff locations in time windows), constraint set C, (environment constraints) and historical decision trajectories up / Dt.
At each decision step I, dispatcher selects an action a, E Ar, where At represents the action space (c.g., assigning taxis to requests and visiting order of each taxi). The system transitions to new state
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Algorithm 2 LLM-Optimizer Interaction Protocol
Require: Population size N, Control interval A, Scenario Information 24
Ensure: Evolved population of prompts and optimized dispatch strategies
1: Initialize population P with N individuals
2: for generation 1 mar mnar_generations do
3:
for round = 1 to Popsize NN do
Can trigger concurrently
4:
parent response, evolution way list GETNEWINDIVDDUA((() Current population
5:
Initialize simulator with environment data En for step &
6:
T do
> Inner simulation loop
7:
E, < GETSIMULATORSTATI way []] = "wayl" then
Current vehicles, requests, etc. Random generation
8:
9:
prompl, <- GENERATER SEPEERATTRADOOPROPP
else if way [1] "way2" then
Memory consideration
11:
prompt, < GENTERITERRVSSSDDROOOPPPUCUU responos response)
12:
else
>> ("way3")
13:
prompt, < GENEEAATEHNOVVATIEEROOOPPPOOOPO response)
14:
end
15:
prompt, prompt, + INNECTCONTEX((4.E))
Add scenario and env data > Call API
16:
response; < QUERYLLM (rropL)
objective < PARSERESPONSEL (response;)
Extract MILP objective
18:
Solve first-level assignment MILP:
19:
min objective
20:
f (ason
Solve second-level sequencing MILP: Can trigger concurrently for different taxis
min Σmar(( pickup 14)
P
23:
X t (UROBISOLVEESeqqeecc, modeel) E11 < UPDATESIMULATOR At)) end for
24:
Evolve At seconds
25:
26:
filness, COMPUTBEIINESSSC
ET)
Total system cost b P. = (Et, objectiie,, filness); for E T
27:
Pnew + P + +P; end for
28:
29:
P +- SORTPOPULATION(P PPP Pnew)
Sort combined population and reserve the top N
end for
S,+1 based on the action is governed by: S f(S,, a.). The search space S for dynamic
T
system is the union of all possible state-action sequences over finite horizon T: S = U [(,,, a,))
==
The action space Ar at each step grows combinatorially with the number of vehicles [V]l and the number of passenger requests |Pr].
For the assignment between passengers and taxis at each step, its space can be ((V))P).
Assume K. requests assigned to each taxi, the number of valid sequences is XCI, with [V] taxis, we can have ((K!))))))
Given steps, the total search space grows as (((()PP))!)))))
This results in an extremely larger search space compared traditional one-time OR problems, making it hard for direct resolution by LLM. While LLM continues to evolve, and current limitations may not persist in future iterations, the complexity the search space necessitates hybrid approach that integrates traditional optimization methods with LLM in hierarchical framework. Therefore, we partition the problem into two levels:
LLM-driven objective design high-level assignment model LLM acts as meta-optimizer, dy- namically proposing adaptive objectives guide decision-making. The generated adaptive objective functions & encoding high-level goals, with current states S, and historical information H,-1 as inputs:
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Each passenger can at most board on one car, as illustrated in min 4 (LM, H4-1) s.t. Σ 1 = 1, ynn E (0.1)
On the one hand, the assignment layer's variables you lack direct visibility into sequencing outcomes (e.g., waiting times, route efficiency), on the other hand, a handcrafted objective (e.g., "minimize assignment distance") is myopic, ignores downstream sequencing impacts. To address this limi- lation, the LLM dynamically generates adaptive objectives D.LAA that implicitly steer assignments toward configurations favorable for low-level sequencing.
Low-level sequencing model
Given fixed assignment plan, each taxi shall solve routing problem to visit the assigned passenger one by one, aiming to minimize the total passenger waiting time:
min (S,, Te)
s.t. g(S,, Te, a4) 0
Where 7r denotes the temporal variables (e.g., taxi arrival time), of is the low-level action, namely, routing variables.
A.6 Direct Modeling Approach
The complete taxi decision-making problem is formulated mixed-logic problem, where the objective to minimize the passenger waiting time, incorporating constraints such as initial car and passenger location/time assignment, task connectivity constraint, vehicle travel dynamics and passenger boarding dynamics. The network we considered is defined as a directed graph G = (S, L}, where S is the of pick-up/dropoff stops in the network, with any stop i S. L is the route path between any two stops. Multiple passengers (taxis) may share the same origins and/or destinations. we utilize the service point sets P ((, O?)), (v,, for origins) and DD (1(p, D²)), ((u, F.)). ). for destinations) to distinguish individual tasks while preserving both passenger geographical information.
Initial and final car location assignment Every time when new planning occurs, the scheduler needs to know the initial taxi location, whether it is in idle state driving on the road this moment, we assume the time left for this taxi to arrive to its next immediate stop shall be known in advance, depicted in constraints (3a) and (3b). respectively. Also, the taxi shall finally go back to the sink location after serving all assigned passengers, as captured in constraint 3c
Σ x S",
(3a)
ŽEP
AR (v,8.)
= 1"
(3b)
Σ 1 Ev) = iEP
(3c)
where wi indicates whether pick-up & dropoff pair ij is selected for taxi w. AR, is the arrival time of car for pick-up service i, where E P.
Passenger assignment constraints
Σ RD
Also, whenever taxi e is assigned to serve passenger it will firstly reach stop OP to pick up the passenger, and subsequently proceed towards stop D to deliver the passenger to its destination. Thus, the operating taxi shall confirm the edge OD for the selected passenger p.
(pp.O)).e..D 2 nd
(5)
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Service connectivity constraints Whenever group of services, each service incorporates pick- up point and dropoff point, are assigned to the taxi, for any two services, the taxi must complete one service before starting the next. In other words, if passengers p and p' are both assigned to the same taxi v, then either passenger p served first or the passenger p' served first. This order is reflected conservation law of typical Vehicle routing problem in current problem formulation, as captured constraint (6).
Vu E
(6a)
j4
TEP
PEP
Fik
(6b)
JED
RED
Also, each passenger must be served by one taxi, as illustrated in 4 accordingly, each pick-up or dropoff service point can only be visited one time by any of the taxi, as described below:
E V.ViEP, PV
ΣΣ
=1
(7a)
ΣΣ JED
dij =1
(7b)
Arrival and departure time constraints If the route edge OD for passenger p is selected for taxi v, then arrival time at dropoff point is equal to the sum of the pick up time at origin point and the OD travel time, Also, departure time dropoff point is larger than its arrival time.
2 (pp.o%)(p.D")) = =1
AR (p. DP. + TROND DP..DM> ARR
(8a)
(8b)
where DP departure time of car v for dropoff service j. where E E. A + B a logic constraint, indicating if A is true, then implies B.
Also, if passenger p' is served immediately after passenger p for taxi v, then arrival time pick-up point for passenger p' is equal to the sum of the departure time of passenger p at its dropoff point and the edge travel time. Also, departure time at pick-up point shall equal to the maximum value between the taxi arrival time and the passenger arrival/request time.
x "(,
AR (P/ ",'r'' = DP" (p. + TR Drow
(9a)
DP "O'O"" (AR; (v orly TP')
(9b)
Objective function The goal is minimize the total passenger waiting time, as captured below:
J= -= WT
(10)
where waiting time WT is obtained from constraints below, only if passenger p is assigned to taxi w, corresponding waiting time WT shall exist, otherwise, should be
n 1 - WT DP (p,00 T 8r
(1
(1lb)
While alternative objectives, such as platform profit or driver earnings, can be incorporated to account for broader stakeholder impacts, this work does not aim to design a comprehensive or multi-faceted objective. Instead, it focuses on the interaction between the LLM and the optimizer, demonstrating how high-level goals specified via natural language can evolve into low-level objectives that guide decision-making Extensions to other objectives are straightforward and left for future work.
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The proposed models above is formulated as mixed logical model, incorporating both linear constraints and logical constraints (8) 9 and (TT). These logical constraints can be equivalently transformed into linear constraints using the big-M method 45 with detailed formulations provided in Appendix A.. Although the problem can reformulated as MILP model and solved directly using commercial solvers such Gurobi to obtain optimal results, its large scale presents significant computational challenges. For instance, a taxi-hailing system, each time can trigger a scenario with 80 taxis and 50 passenger requests, the decision variables zi alone result in 200,000 (80$50#50) variables MILP. Given the short decision-making intervals required in real-time operations, solving such a large-scale MILP optimally within the available time frame is impractical. Therefore, the problem is typically formulated as a sequential approach: first, solving an assignment problem to allocate subset of passengers group of taxi drivers, followed by solving traveling salesman problem with additional time constraints for each taxi independently in parallel.
A.7 Logic Constraints Conversion
In Section A.6, the problem has been formulated as a mixed logical dynamic model, which involves logic constraints, e.g. e.g.varrival and departure time constraints (8). (9). and waiting time constraints (TI). All these logic constraints can be converted to linear constrants via big-M method as follows.
Proposition 1 Replacing logic constraints ( with Inequalities solution.
12 in the model leads to the same
Proof: Let M1 be sufficiently large, and satisfies M1 2 max max{=(AR) (p. DP (p.or) TR Dr)), (AR (n.rr) - DPY (. (.)) for E V,P E P), then constraint (8) can be rewritten as:
Yo E E.PPP.
(n.D - DPP (p.0v) TR(OrDD S M1(1 - 2 (0.00)6.
(12a)
AR. D') + DPp O'RR(Or,D')) Dr) M1(1 xy
(12b)
D. DP (p.Do) S MI y ((p.Or)p.DDm)
(12c)
AR (p.Dr) + DP (p. S [1 (pp.rr(p.DD) (p.Dr)
(12d)
Proposition 2 Replacing logic constraints p with Inequalities 13) in the model leads to the same
solution.
Proof: Let M2 be sufficiently large, and satisfies M2 2 max ( (oo) DP (p..D.) (p.Dr) DP Wi To' - DDP D w,, for E V,P E P}, then constraint
TR(D=, "o)),,or)),A) AR. { can be rewritten as:
(p',or') (
Vu E E
AR" (w,0 DP (p.Do) - TRR S Mp(1 x (n,
(13a)
ARR Ao + DPP (p. + +TR ,00) I M2(1 IY I(
(13b)
DP (g' DD') - AR (/.. 0P') 2 21 x (p.E")('' ,D'r))
(13c)
DP Grr - -IV' 2 -M2(1 x (pp
(13d)
Proposition 3 Replacing logic constraints 17 with Inequalities solution.
14 4 in the model leads to the same
Proof:
Let Mj be sufficiently large, and satisfies M₃ mar(=(WTT" DP (p..rr)
((WT")) for E V.P E P}, then constraint (11) can be rewritten as:
Vr E V.PEP WT - DP..or) T M₂(1 )0 W/Te +DP + T < M3 )0
(14a)
(14b)
We < May
(14c)
WT 2 mM
(14d)
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A.8 Dynamic Simulator Environment Setup
In the simulator, the state of the traffic network at time instant is simplified as y(k) + (k), E I), where is the index of the taxi, I is the set of all taxis, d,(k) is the lo- cation the i-th taxi at time instant is travelling to, and (("(k) is the time of the i-th taxi arriving that location. The following conventions are made for state and task update in the simulation:
I): the task being executed will not be reassigned.
2): if taxi has no task assigned, it will stay at destination of the last task.
The command issued from optimizer is set of task sequences for each taxi in the traffic sys- tem. Each task has four parameters: origin o, destination d, passenger arrival time (stnt taxi departure time tdep, We denote the command for the i-th taxi from the optimizer at time instant k as opt (k) and the command being executed as (k).. When the command from optimizer is issued to taxis, the command u, (k) will be set as u (k). The command (k(k) is defined as u. (k(k) = (task,,o(k) ((k),,,((k...................... ...)...., t (k)), where (k) (k),( (k)), 27, (k), tig (k),,e,) dep (k)) collecting origin, destination, taxi arrival time, passenger arrival time and taxi departure time from task origin, respectively. With system state y(k) and command u(k) (k, Z), the state is updated as y(k 1) f(v(k), u(k), k) where f(,,,,)) is given as in Algorithm for each k. When receiving new computed oo (k) {ur"
opt (k)) E Z}. the state and command shall be updated as in
Algorithm 3 State Transition Model
Input: Current state y(k),, commands u(k)
2: Output: Next state y(k +
3: & + ++
4: for each taxi E I do
5: (k:) + then
(Oc, de, tar usturt dep + (u;())
6:
7:
if k f.th then
8:
if' d, = then
9:
Update destination: d; + de Update arrival time: 1.th +- ddep TR
11:
else
12:
(k)(
13:
if u. (k) + then
(Oc, de, thrt estart Eder) (k))(u;(k)) Move to next task: thth + maxik, tsart) + T TRd di +
15:
end if
17:
end if
end
19: else
Mark taxi as idle
21: end if
22:
end for
Algorithm 4 Command Update Protocol
1: Input: Optimized commands (k)), current state y(k)
2: Output: Updated commands u(k: + 1)
3: for each taxi E I do
4: if' u, = 9 then
5:
u,(k + 1) + (k)
6: else
7:
Merge commands: (k + 1) +- (opeend(u(H)) u4 (1))
8:
end if
9:
end for
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A.9
Prompt Engineering
In this section, we provide the details of prompt strategies used our method.
A.9.1
Scenario prompt compositions
Figure [5 illustrates the scenario prompt structure introduced in section 3.2. where the scenario prompt is composed by Psys, Pgeo and P model, respectively.
Scanario Prompt:
Geospatial context
Act se All transportation on aptimization expert designing the iimrrry unjuctive function
OD Explanation
real- hal-time mooiitty system
(Origina)
System Overview
Optimization Structure
Values (Destinctions)
Va Tronel swoonds between enign (row) destination (oxlumn))
Simulator jesecutes
commanei)
Dispotcher
Firat Levet: Passenger-tasd
Jonécuratiin strategies) closed lop
assignment problem
First- model blusprint
aequencing
locationsl,
Passagge c4ss
Structured Explanation
passenger
minimize passenger
core components:
stop offs)
requestalppick
Texi data closs, Include start_pos, ortvvl time, and tasi index:
Passengan date class, include origin, destiration, antival time, and possenger Inde
Your Taak
Requirements
Design novel first- level objective funesion usings
Enoble timely service for all pessargera
cofladd tor porry
Assignment Mocel: Gurobi-basse
optinization engine
dey, mejhodss
Stot' map data
Compatible with Gurati-based MIL teruraniin
birary soeigmment
variublis yEs
request details
Balance ussianment etficiency
axaigned reactly onw taci
setupCone: Ensures each
SetupOaj: Cenfigrablee onjective function
feasibility
Figure 15: Scenario prompt structure provided to LLM, including system architecture and role defintion (left panel), geospatical content (upper right panel), first-layer model blueprint (lower right panel)
System architecture & role definition (Pays) The system architecture and role definition establish the LLM as an adaptive objective designer rather than an end-to-end solver.
System Architecture: We provide a brief overview of the system architecture, including the commu- nication between the simulator and the dispatcher, the two-level decomposition of the dispatching problem, namely, Massign min, Lussinn (y) (LLM) and Mroute min, Croute (xly").
Role Specification: Define the LLM'ss role constrain its outputs to objective function components compatible with the Gurobi 42 solver API using template-base -baed generation.
Geospatial context Pgeo As the problem is related to ride-hailing services, incorporating geospatial context is essential. Therefore, we provide the testing environment details to the LLM. Utilizing the publicly available New York or Chicago taxi dataset, our analysis focuses on 19 zones in either Downtown Manhattan extended Central Chicago. Static map data is represented matrix WoD, encoding spatial semantics via the Manhattan zone graph = (V, E, WoD) where Wo = to Rt, V(u,, Uy)
an OD
First-layer model blueprint Pmodee The first-level assignment model class structure is exposed to guide L.M--copatible objective design. This class structure provides framework for the LLM to work within, ensuring that its output is consistent with the overall model design. By having access to the class structure, the L.M can understand the relationships between different variables and design an objective function that takes these relationships into account.
36

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A.9.2 Harmony search prompt instruction
As discussed in section 3.2.3 evolutionary operators employ structured instruction blocks to guide LLM-based prompt optimization. Specific instructions are listed as follows:
Heuristic Improvement (W2) Instruction Block: Focuses on incremental objective refinement:
(a) Temporal alignment: Incorporate taxi-passenger arrival time coordination.
(b) Resource weighting: Adaptive taxi utilization coefficients
(c) Structural preservation: Maintain 50% legacy objective components.
Innovative Generation (W3) Instruction Block: Promotes paradigm-level innovations:
(a) Emphasis goals: design first-level decisions Vr.p minimize expected second-level waiting:
max (DP( -0) |Be
min
E
PEP
(b) Multi-horizon optimization: Joint current/future state consideration
(c) Dynamic weight adaptation: Time-varying priority coefficients
A.9.3 Gurobi format restriction
To ensure the objective function compatible with Gurobi, additional constraints introduced enhance its feasibility. Two types of restrictions are imposed: 1. Admissible Terms: These constraints define the complete structure of the objective function, ensuring all components in the function adhere to the class requirements. valid operations and transformations permitted in Gurobi's expression framework maintain solver compatibility.
Expression Construction Rules: Guidelines specifying the
Admissible components
The generated objective functions must utilize only the following class elements, which are also summarized Table
Decision variables: Binary assignments: self y[v,p) E {0, 13
Static parameters:
Distance matrix: self [o] [a]
Big-M constant: self.M
Taxi state (read-only):
Current position: self. taxi [v] starr__os
Availability time: self taxf.taxiff [v] arrivallttiiii time
Passenger state (read-only):
Origin/Destination: self passenger .oiigin self passenger [p] destination Request time: self fallf.aasseggerffn [p] arrTime
Table 7: Model specification mapping
Concept
Mathematical Form
Code Implementation
Assignment variable
Ve.p
self yolf.y[v.. [v, self. Sil.dditMattiix
Travel time matrix
TR,
Taxi availability
T
self. self.taxi [v] arrival_tii
Passenger request time
Fog
self pass.aassengerB [p] arrTime
Weighted sum
Σ wics
sum(w * c for c in zip(veights, costa))
37

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Expression construction rules All objectives must adhere to:
1. Structural requirements:
Costs stored in list costs (1-5 elements)
Weights in list weights (length matching costs) Final objective: sum (w*c for c,c in eip(weghhs,, costs)
2. Gurobi expression rules:
Use **gb. **gb.quickes quuckummm () //gb. max_ //gbb abs_ ()** ** only when containing variables
Use **sum
(0mmmmm/////x /max () /abs () ** when working with parameters
Quadratic terms via variable multiplication (y [v.p]+y (y, [v,g])
Never multiply Gurobi variables with Gurobi expressions
Invalid: y y[v gb. max_ (expression with_warrtht 0)
Never use Python if/else with Gurobi variablee/exppressions
Generation protocol
The following template is provided to L.LM for valid objective construction:
def
cost1 [Proper Gurobi expression]
cost2 = [Proper Gurobi expression] # Add more components as needed
# Custom weights (match costs length) weights
[Your custom weight 1], [Your custom weight 2], # Add matching weights
objective = sum(wec for w,c in zip costs)) self model. setObjective (objective, gb......M..... (RB.MIIIMIZE)
A.9.4 LLM response examples
Figure 17 and Figure 18 illustrate four L.LM response examples generated by the proposed method. The first-layer model blueprint defines the default objective function, serving as a foundational heuristic for the LLM, as captured in Figure 16 The listed four objective function variants capture different optimization strategies:
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Figure 16: Default objective code used in first-layer model blueprint
Weighted multi-objective with dynamic penalties As captured in left panel of Figure 17 unlike the static weights in the default, this variant introduces dynamic penalties (c.g., scaled idle taxi prioritization in cost2, future availability adjustments costs) and amplifies priority weights (e.g.. 80 for waiting time, 1000 for load imbalance). By dynamically scaling penalties based real-time conditions (e.g., taxi arrival vs. passenger request times), adaptively balances short-term efficiency and long-term fleet availability, addressing the default's rigidity in handling time-dependent scenarios.
Quadratic load balancing with reassignment penalty While the default penalizes load imbalance via costa, this variant (shown in right panel Figure [7) explicitly enforces faimess through a squared deviation from the average passenger-per-taxi ratio (load_ (ooa_balancce) Furthermore, this variant incorpo- rates a quadratic reassignment penalty (cosl3) to minimize frequent taxi reassignments. This dual focus effectively reduces reassignment churn and strengthens fairness, addressing the limitations of the default model in enforcing balanced allocations.
38

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Busy taxi penalty with load balancing Replacing the default's 's simplistic waiting time penalty, this variant (captured in left panel of Figure (8) directly penalizes assignments busy taxis using operator (cos(4) to check if a taxi's availability precedes the passenger's request. This variant incorporates the full trip duration into cost2; extending beyond the default model's focus on pickup times alone. Additionally, it maintains quadratic load balancing to distribute demand more effectively. By considering both real-time availability and total trip effort, this approach prevents excessive assignments to already occupied taxis, addressing a key limitation of the default model.
Sequential assignment with dropoff-pickup chaining Unlike the default's isolated trip modeling, this variant (captured in right panel of Figure [8] introduces sequential chaining mechanism in cost3, penalizing sropoff-to-pickine travel time between passengers (y(v, (V, pl y(u, P_D. also adds bidirectional waiting time penalties (costi for taxi delay, costs for passenger earliness), whereas the default only penalizes taxi delays. By optimizing trip sequences and accounting for both waiting constraints, it enhances fleet utilization efficiency, which is improvement single-trip optimization approach.
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Figure 17: LLM response examples, including dynamic penalty variant (left panel) and quadratic loading balancing variant (right panel)
39

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Figure 18: LLM response examples, including busy taxi variant (left panel) and sequential assignment variant (right panel)
40


Reviews:
[{"review_id": "Gq4nVMCp7H", "reviewer": "Reviewer_E53E", "summary": "This paper introduces a hybrid framework that combines large language models (LLMs) with mathematical optimization to address dynamic decision-making, specifically for mobility-on-demand systems. The approach uses pre-trained LLMs to generate high-level objectives and employs mathematical optimization for detailed dispatching strategies, creating a system that doesn't require extensive training.\n\nThe method integrates LLM-driven semantic reasoning with solver-enforced mathematical rigor, dynamically updating objectives through prompts. This closed-loop mechanism balances exploration and feasibility, achieving state-of-the-art performance on urban mobility benchmarks. Experiments on New York and Chicago taxi datasets demonstrate the system's effectiveness in minimizing delays and maintaining spatial consistency. Detailed analyses, including ablation studies and hyperparameter sensitivity, provide comprehensive evaluation of the framework's components.\n\nOverall, the paper presents a creative integration of NLP and optimization methods, offering a new approach to complex real-time decision-making. By showcasing adaptability across different urban environments and demand patterns, the authors highlight practical applications and future research potential in combining data-driven priors with mathematical optimization for robust decision-making systems. The thorough experimental setup ensures reproducibility and extension of this promising approach.", "questions": "1、How do you plan to incorporate more dynamic factors, such as traffic congestion or road disruptions into your model? Could you elaborate on potential strategies or experiments that might address these limitations?\n\n2、Can you provide a deeper comparison with existing state-of-the-art techniques, highlighting both the advantages and limitations of your approach relative to others?\n\n3、Could you explain further how the LLM adapts through prompt-level feedback? Are there any plans to integrate reinforcement learning techniques to improve adaptability?", "limitations": "The authors have partially addressed the limitations of their work, particularly regarding technical constraints such as the static adaptation mechanism of the LLM and the lack of fine-grained traffic dynamics in the current experimental setup. However, they have not sufficiently discussed potential negative societal impacts or broader ethical considerations associated with deploying such a system in real-world, high-stakes environments.\n\nFor instance, the use of LLM-generated prompts in decision-making systems could introduce biases or unintended consequences, especially if the generated objectives inadvertently prioritize certain user groups or regions over others. Additionally, overreliance on automated dispatching systems without proper human oversight may lead to safety concerns or reduced job control for drivers in mobility-on-demand platforms.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths：\nThe paper stands out for its high quality and clarity, offering a well-documented experimental setup that ensures reproducibility. The integration of large language models with mathematical optimization techniques is clearly explained, making the methodology accessible to both experts and non-experts. By demonstrating practical applications on real-world datasets from New York and Chicago taxi services, the authors highlight the significant potential of their approach in urban mobility systems. This combination not only enhances the accuracy of predictions but also provides confidence intervals crucial for decision support systems.\n\nWeaknesses：\nDespite its strengths, the paper has some limitations, particularly regarding the applicability of its model in more realistic operational conditions. The current setup does not fully account for dynamic factors such as traffic congestion or road disruptions, which could limit the model's effectiveness in real-world scenarios. Additionally, while the proposed method is innovative, a deeper comparison with existing state-of-the-art techniques is lacking, leaving gaps in understanding its unique advantages and potential areas for improvement. Addressing these aspects would strengthen the paper’s overall impact and broaden its applicability.", "was_revised": true, "final_justification": "Thank you for your detailed response, and most of my concerns have been solved.", "rebuttal": "**We appreciate the reviewers’ thorough feedback. We have grouped both the identified weaknesses, questions and limitations into the following comments for clarity and response.**\n\n**Comment 1**. How do you plan to incorporate more dynamic factors, such as traffic congestion or road disruptions into your model? Could you elaborate on potential strategies or experiments that might address these limitations?\n    \n**Response**: Thanks for the reviewer's comments. Our work establishes a hybrid LLM $+$ optimizer framework that combines high-level reasoning with numerical optimization for dynamic decision-making. The LLM serves as a meta-optimizer to evolve high-level objectives, while a mathematical solver ensures feasibility and rigor. Experiments conducted under New York and Chicago datasets achieve an average reduction of 16% on passenger waiting time, demonstrating that our approach can adapt to different environment dynamics.\n\nTo incorporate more dynamic factors and make the test scenarios even more realistic, we plan to extend our framework along the directions below:\n        \n**Enhanced Traffic Simulation**: Our current setup already uses real-world traffic data to derive empirical distributions for realistic taxi locations and passenger request times, capturing the stochastic nature of urban mobility. Future work will further enrich the simulation by incorporating broader traffic patterns, including non-taxi vehicles and junction signals.\n        \n**What-if Scenario Analysis**: We will conduct experiments to assess how the proposed method responds to unexpected events, such as accidents, road blockages, and others. This stress testing will help evaluate robustness of the LLM-guided optimization process.\n\n**Utilization of Advanced Traffic Simulator**: We will replace our current simulator class with established traffic simulation platforms like SUMO or VISSIM, which support dynamic vehicle interactions, signal control, and detailed road network modeling.\n\nWe believe that these improvements, aimed at incorporating more dynamic elements, will allow us to simulate complex urban scenarios with realistic congestion patterns and road disruptions. Therefore, we can gain insight into how AI-generated targets perform in dynamic, uncertain transportation environments.\n\n**Comment 2**. Can you provide a deeper comparison with existing state-of-the-art techniques, highlighting both the advantages and limitations of your approach relative to others?\n\n**Response**: Thanks for the reviewer's comments. We provide the advantages and limitations of our hybrid LLM$+$optimizer approach against state-of-the-art baselines as follows, supported by empirical results from Tables 2 and 5, and spatiotemporal analysis (Appendix A.1.1) in the paper.\n\n**Methodological Advantages of Our Hybrid Approach**: \n- To the best of our knowledge, this is the **first work to combine LLM-driven semantic reasoning with classical mathematical optimization** in a closed-loop framework for sequential, dynamic decision-making.\n- Unlike existing baselines, such as manually-designed objective functions, RL-based methods, or pure LLM-guided solvers (e.g., FunSearch, EoH), our framework is **training free**, and allows the **LLM to generate high-level objectives** via prompt-based reasoning, while a **solver guarantees constraint satisfaction and numerical rigor**.\n- We propose a **novel harmony search algorithm** with three operators (random inference, heuristic improvement, innovative generation) that iteratively refine LLM-generated objectives based on solver feedback. \n\nThis mechanism of LLM-guided objective refinement through solver feedback is not present in any state-of-the-art approaches.\n\n**Empirical Advantages of Our Hybrid Approach**: \n- *Superior Scalability and Solution Consistency*: Manual designed objectives (e.g., weighted distance-temporal-utilization terms) suffer from significant performance degradation at scale, e.g., 9.27 min for case 9 in Manhattan scenario in Table 2. However, our approach consistently achieves ~40\\% lower waiting times across large-scale scenarios, e.g., Ours: 4.01 min vs. RL: 6.82 min (best state-of-the-art) for case 9 in Table 2. \n\n- *Robustness to Spatiotemporal Heterogeneity*: Manual objectives, RL and LLM-only methods fail in imbalanced distributions, e.g., Chicago case 9, RL yields 20.03 min delay in Table 5. However, our approach reduces waiting times by ~20\\% in Chicago’s large-scale, unbalanced zones, e.g., case 9: 15.37 min vs. EoH: 19.79 min. \n\n **Current Limitations**: \n\n- *Low-Demand Period Overhead*: During low-demand periods (e.g., nighttime), our system still operates with full objective generation and solver cycles, which may be unnecessary. This can be mitigated by time-segmented deployment, using simpler objectives during off-peak hours, while applying our full method during peak periods.\n\n- *LLM Infrastructure Requirements*: Compared to rule-based methods, our method requires LLM infrastructure and proper prompt engineering. However, this cost becomes negligible in large-scale deployments and can be further reduced by using locally hosted models or lightweight techniques.\n\n**Comment 3**. Could you explain further how the LLM adapts through prompt-level feedback? Are there any plans to integrate reinforcement learning techniques to improve adaptability?\n\n**Response**: Thanks for the reviewer's comments. \n\n**How the LLM adapts through prompt-level feedback**\n\nOur approach integrates Harmony Search (HS), multi-query inference, and dynamic state streaming to facilitate real-time LLM adaptation within a closed-loop control framework. \n\nAt each simulation timestep $t$, we execute closed-loop prompting, where the LLM receives structured feedback reflecting the current system state and generates updated dispatch objectives accordingly. The dynamic prompt $\\mathcal{P}_{\\text{dyn}}$ includes two core components:\n    \n- *Vehicle states*: $\\text{veh}_t \\in \\mathbb{R}^{|\\text{veh}| \\times 2}$, real-time positions and expected arrival times.\n- *Passenger demand tensor*: $\\text{pass}_t \\in \\mathbb{R}^{|\\text{pass}| \\times 3}$, origin, destination, and request timestamps.\n    \nHS is used to optimize prompt structures across generations. Multi-query inference enables iterative refinement based on optimizer feedback. Dynamic prompting ensures the LLM remains contextually grounded in the evolving simulation environment. Together, these components form a co-adaptive feedback loop between the LLM and the optimization process.\n    \nThis integrated mechanism is validated in Table 4 and Figure 9 in the paper. In particular, we compare multi-query (closed-loop) against single-query (open-loop) strategies. The results show that multi-query consistently achieves up to **50\\% lower cost** within 10 iterations. Notably, the steepest cost reductions occur in the early iterations, demonstrating the efficacy of prompt-level adaptation in accelerating convergence.\n\n**RL-based plan**\n    \nWe agree this is a worthwhile direction and is a potential extension to our framework. Unlike the baseline RL methods evaluated in our paper, this extension will emphasize the co-adaptation between the LLM and the optimizer for enhanced performance.\n\nWhile baseline RL methods directly learn low-level dispatch or routing policies, our planned RL integration treats the RL algorithm as a tuner that improves the LLM’s ability to generate high-level, interpretable objective functions using feedback from the optimizer as a critic. Specifically, \n- *LLM* acts as *actor* to generate objective functions.\n- *Optimizer* acts as *critic* to evaluate objectives by solving a mixed-integer programming problem. The output $W$ of the optimizer is used to update RL rewards. \n- *Reward*: $\\mathcal{R} = -\\Delta W + \\lambda \\cdot \\text{Explainability Score}$, where $\\Delta W$ is the negative change in cost (e.g., passenger waiting time, $W_{k+1} - W_{k}$) and $\\lambda \\cdot \\text{ExplainabilityScore}$ is the weighted human rating, enforcing human alignment.\n        \nGiven the effective performance of the method proposed in this paper, it can be considered as a natural baseline for this RL-based extensions, meanwhile eliminating the need for time-consuming dataset management and extensive fine-tuning.\n\n**Comment 4**. Limitations. The authors have not sufficiently discussed potential negative societal impacts or broader ethical considerations associated with deploying such a system in real-world, high-stakes environments. Additionally, overreliance on automated dispatching systems without proper human oversight may lead to safety concerns or reduced job control for drivers in mobility-on-demand platforms.\n\n**Response**: Thanks for the reviewer's comments. We acknowledge that automated decision-making systems must be designed carefully to avoid unintended biases. While our current focus is on optimizing measurable metrics such as waiting time, detour time, and load balance, we agree that incorporating fairness or bias mitigation constraints is a valuable future direction. \n\nOur framework is extensible and can incorporate fairness-aware objectives or support post-hoc audits to identify and address potential disparate impacts. In contrast to fully autonomous systems, our method follows a co-optimization paradigm: the LLM generates high-level objectives, and a solver ensures feasibility and handles execution. This design facilitates human-in-the-loop control and allows domain-specific constraints to be directly embedded in the optimization model.\n    \nWe believe early limitations should not hinder forward-looking research. Agentic AI has demonstrated significant progress in many domains, and holds strong potential for intelligent transportation. Our system’s ability to dynamically learn and adapt dispatch strategies offers clear social benefits, reducing passenger wait times (**~16% improvement**), boosting efficiency, and easing city-wide congestion."}, {"review_id": "euKRQhHI2R", "reviewer": "Reviewer_7QZG", "summary": "This paper proposes a training-free hierarchical optimization framework for mobility-on-demand systems, where a large language model (LLM) dynamically generates high-level assignment objectives. These are refined via a harmony search-based evolutionary loop and solved with a low-level optimizer for routing under constraints. Experiments on real-world taxi datasets show significant improvements over RL and LLM-only baselines.", "questions": "1. Can authors provide more clear movitation for selecting HS, instead of other ES? (see Cons 1)\n2. Can authors provide experiment results over large scale of taxis and passengers, for example, thousands of taxis? (see Cons 2)", "limitations": "\"yes\"", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n\n1. The integration of LLMs and optimization in this paper is novel. \n2. The training-free scheme is practical, which requires no training data comprared with RL-based approaches.\n3. The proposed approach is evaluated on large-scale, real-world taxi datasets, demonstrating consistent and significant improvements over SOTAs.\n\n\n\nCons:\n\nMy major concerns include **the introduction of HS**, and **the scalability of method**.\n\n1. The motivation for using HS is weak, and the paper does not explore alternative evolutionary strategies to justify this choice.\n2. The problem scale seems unrealistic, as real-world ride-hailing systems typically involve thousands of taxis, while the experiments are limited to hundreds.\n3. The framework relies on frequent LLM queries, but the runtime cost and inference latency in real-time deployment are not analyzed.\n4. Figure 1 is overly complex and unclear. the difference between WAY 2 and WAY 3 is not visually or conceptually well-distinguished.", "was_revised": true, "final_justification": "I have thoroughly read the rebuttals from the reviewers, and part of my concerns have been resolved, and I will keep my score", "rebuttal": "**We appreciate the reviewers’ thorough feedback. We have grouped both the identified weaknesses and questions into the following comments for clarity and response.**\n\n**Comment 1**. The motivation for using HS is weak, and the paper does not explore alternative evolutionary strategies to justify this choice. Can authors provide more clear motivation for selecting HS, instead of other ES? \n\n**Response**: Thanks for the reviewer's comments. Our selection of HS is grounded in the unique challenges of evolving LLM prompts, which encode rich problem context, including environment state $S_t$ and historical llm-generated objectives $H_t$ in natural language. Meanwhile, traditional evolutionary strategies, such as Genetic Algorithms (GA), typically rely on crossover and mutation operations at the individual solution level. These operations suffer from several fundamental limitations when working with LLM prompts:\n\n1. ***Loss of Coherence***: GA’s crossover operation between arbitrary prompt segments often results in semantically inconsistent text, making it difficult for the LLM to understand or generate meaningful responses.\n```\nIn contrast, the HS algorithm operates on complete prompts using operators that preserve natural language structure. It either generates new prompts from scratch or reuses full parent prompts, avoiding disruptive operations like crossover and mutation.\n```\n2. ***Fitness Inconsistency***: In our case, GA’s crossover and mutation operations alter the structure of a prompt, breaking the link between its phrasing and the LLM's response quality. More specifically, modifying LLM-generated objective functions (e.g., weights or costs) disrupts the alignment between prompts and their fitness scores, making parent fitness values unreliable for guiding future search.\n```\nHS avoids this issue by either generating new prompts without relying on prior fitness scores to promote diversity, or directly reusing full parent prompts with their known fitness values.\n```\n3. ***Parsing Complexity***: To perform crossover and mutation correctly, additional parsing logic is required to identify and recombine meaningful prompt segments, increasing implementation complexity and reducing algorithmic robustness.\n```\nSince HS does not use crossover or mutation, it eliminates the need for prompt parsing and maintains a simpler, more robust design.\n```\nTo summarize, HS is selected due to its advantages on maintaining the structural and semantic constraints, which is required for stable LLM interaction.\n\n**Comment 2**. The problem scale seems unrealistic, as real-world ride-hailing systems typically involve thousands of taxis, while the experiments are limited to hundreds. Can authors provide experiment results over large scale of taxis and passengers, for example, thousands of taxis? \n\n**Response**: Thanks for the reviewer's comments. We conducted additional large-scale experiments that extend our previous settings. Specifically, we scaled our simulations to include up to 1000 taxis and 1000 passengers within the Manhattan and Chicago environments, under longer simulation durations.\n\n    The results are summarized in Table I. Consistent with the small-scale findings in Tables 2 and 5 in the paper, the Manhattan scenarios in large cases below consistently exhibit lower average passenger wait times compared to Chicago, owing to their more spatially balanced demand distribution. Conversely, the more skewed distribution in the Chicago scenarios leads to longer delays under high demand levels. These results demonstrate that our method maintains solution quality and computational tractability even at significantly larger problem scales.\n\nTable I. Experiment for scenarios under 1hour simulation period\n| Scenarios | Manhattan | Manhattan | Manhattan | Manhattan | Chicago | Chicago | Chicago | Chicago | \n|--------|--------|--------|--------|--------|--------|--------|--------|--------|\n| Taxi No. | 300 | 800 | 1000 | 500 | 300 | 800 | 1000 | 500 |\n| Passenger No. | 500 | 500 | 800 | 1000 | 500 | 500 | 800 | 1000 |\n| Avg. Wait time (min) | 3.69 | 0.47 | 1.82 | 4.64 | 5.21 | 1.02 | 13.65 | 6.23 | \n\nMoreover, for real-world deployments involving tens of thousands of agents, our framework can be extended through a distributed architecture. In such settings, multiple instances of our controller can operate in parallel across different city zones, with coordination mechanisms ensuring global consistency and service quality. \n\n**Comment 3**. The framework relies on frequent LLM queries, but the runtime cost and inference latency in real-time deployment are not analyzed.\n\n**Response**: Thanks for the reviewer's comments. We have provided the actual runtime statistics for the optimization & prompt evolution components across the Manhattan and Chicago cases.\n```\nAs shown in Table II and Table III, although runtime increases with the simulation scale (e.g., longer durations and more demand), it remains substantially lower than the simulation time itself. For instance, Case 9 in Table II, which involves a 1200s (20min) simulation, completes in less than 4min, even though it includes multiple rounds of optimization. This demonstrates that the framework can operate within practical time constraints.\n```\nTable II. Run time (s) for Manhattan cases \n| Sim Duration (s) | 600 | 600 | 600 | 900 | 900 | 900 | 1200 | 1200 | 1200 |\n|--------|--------|--------|--------|--------|--------|--------|--------|--------|--------|\n| Cases | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 | Case 9 |\n| Run Time (s) | 127.27| 138.89 | 107.33 | 178.89 | 194.51 | 178.43 | 233.50 | 208.70 | 196.33 |\n```\nSimilar to Manhattan case,  Chicago case (Table III below) run times grow moderately with the problem scale, yet all scenarios are solved within 4 minutes, demonstrating practical feasibility. \n```\nTable III. Run time (s) for Chicago cases \n| Sim Duration (s) | 600 | 600 | 600 | 900 | 900 | 900 | 1200 | 1200 | 1200 |\n|--------|--------|--------|--------|--------|--------|--------|--------|--------|--------|\n| Cases | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 | Case 9 |\n| Run Time (s) | 119.31 | 133.19 | 116.82 | 141.21 | 149.80 | 178.70 | 223.89 | 155.63 | 216.92 |\n\n**Comment 4**. Figure 1 is overly complex and unclear. the difference between WAY 2 and WAY 3 is not visually or conceptually well-distinguished.\n\n**Response**: Thanks for the reviewer's comments. We appreciate the reviewer’s observation regarding the lack of clarity in differentiating WAY 2 and WAY 3 in Figure 1. The original design aimed to present a unified framework encompassing three evolutionary strategies: \n- WAY 1: pure exploration from scratch. \n- WAY 2: exploitation via incremental refinement of elite prompts. \n- WAY 3: exploitation via structural recombination and reinvention.\n\nWhile both WAY 2 and WAY 3 share downstream optimization components to emphasize architectural cohesion, the visual simplification unintentionally obscured their conceptual distinctions. To address this, we have revised Figure 1 with the following modifications: \n- *Visual Differentiation*: Label WAY 2 as “Incremental Refinement”, using a red block with dashed borders to denote constraint-aware local adjustments. Label WAY 3 as “Paradigm Innovation”, using a green block with double borders to signal structural novelty via recombination. \n- *Instruction Mapping*: The figure should clearly include the main prompt-level mechanisms. For WAY 2, these are temporal alignment, resource weighting, and preserving partial existing parent semantics. For WAY 3, the focus is on long-horizon impacts, increasing the use of idle taxis and innovative goals.\n- *Structural Simplification*: Collapse shared modules into a single “Shared Optimization Layer” to reduce visual redundancy. Differentiate data flow: solid arrows for WAY 2, zig-zag for WAY 3."}, {"review_id": "Lckz2bITuO", "reviewer": "Reviewer_ap7C", "summary": "This paper proposes an LLM-integrated hierarchical optimization framework for mobility-on-demand systems. Specifically, an LLM is employed to generate the objective function for the assignment problem, where the harmony search algorithm iteratively refines the generation prompts. The optimization problems at the first and second levels are then solved numerically. The proposed framework achieves higher performance compared to baseline approaches, including methods based on manually designed objectives, RL-based methods, and purely LLM-based methods, when evaluated on real-world taxi datasets.", "questions": "- Could smaller language models be fine-tuned to generate effective objective functions, even with fixed prompts? What is the main advantage of the proposed evolutionary framework compared to fine-tuning?\n- Would it be impossible to design effective objective functions using rule-based methods? Is the use of LLMs necessary for generating high-quality objectives? How complex are the behaviors demonstrated by LLMs in the experiments?", "limitations": "Yes.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strengths:**\n- This work proposes a novel method for integrating LLMs with optimization problems that can leverage any pre-trained LLM. There is considerable room to further enhance the proposed framework by fine-tuning the LLM.\n- The proposed framework is described clearly and thoroughly, making it easy to reproduce.\n- The empirical evaluations are comprehensive and promising, demonstrating that the proposed framework generally outperforms the baseline methods.\n\n**Weaknesses:**\n- The proposed algorithm relies on multiple heuristics, making it challenging to clearly evaluate how each component individually contributes to the overall performance.", "was_revised": true, "final_justification": "While the motivation behind the harmony search and each component of the framework is not very clear, and the scalability of the algorithm has not been clearly demonstrated, I think the proposed framework is promising and a meaningful contribution. I will keep my score.", "rebuttal": "**We appreciate the reviewers’ thorough feedback. We have grouped both the identified weaknesses and questions into the following comments for clarity and response.**\n\n**Comment 1**. The proposed algorithm relies on multiple heuristics, making it challenging to clearly evaluate how each component individually contributes to the overall performance.\n    \n**Response**: Thanks for the reviewer's comments. We appreciate the reviewer’s insightful observation about the challenge of individual component contributions. \n\nThe integrated design is essential to address the inherent complexity of dynamic ride-hailing dispatch. Specifically, the environment is highly stochastic, partially observable, and multi-objective in nature, which makes end-to-end training or rule-based optimization alone insufficient. A hierarchical approach is thus required, where separate components handle specific challenges but operate together seamlessly: \n- The **high-level module**, aided by the *LLM*, is responsible for generating interpretable and flexible dispatch objectives that adapt to evolving traffic and passenger patterns. \n- The **low-level routing solver** ensures that these objectives are translated into executable dispatch plans that satisfy hard constraints. \n- The **harmony search loop** acts as a *meta-heuristic* that iteratively refines the prompts to query LLM using simulation feedback, driving the co-adaptation between the semantic layer and the optimization layer.\n\nOur system modules fit together seamlessly, and modules are interdependent. For example, the LLM alone cannot account for downstream feasibility without the routing feedback, and the routing solver depends on the quality of high-level objectives to make meaningful decisions. In this sense, the LLM and harmony search together form a co-evolving meta-heuristic, and isolating them would undermine the core mechanism of design.\n\nWe believe our empirical results offer strong indirect evidence for the effectiveness of each component. Extensive experiments across diverse scenarios based on both New York and Chicago taxi datasets, demonstrate **an average reduction of 16% on passenger waiting times** over state-of-the-art baselines (see Table 2 and Table 5 in the paper). Specifically,\n- In Manhattan testing, our approach consistently achieves ~40\\% lower waiting times across large-scale scenarios, e.g., for Case 9 in Table 2 in the paper, our approach is 4.01 min compared to RL’s 6.82 min.\n- Under large demand case, FunSearch completes all pickups in 5–7 slots in Manhattan and 6 slots in Chicago, while RL and manual baselines need 8/9 slots in the latter (see Figures 7 and 8 in the paper Appendix). However, our method consistently fulfills all requests in just 4 slots (2400s in Manhattan, 3200s in Chicago), achieving 40–50\\% faster completion.\n\n**Comment 2**. Could smaller language models be fine-tuned to generate effective objective functions, even with fixed prompts? What is the main advantage of the proposed evolutionary framework compared to fine-tuning?\n    \n**Response**: Thanks for the reviewer's comments. While fine-tuning smaller LLMs for objective generation is possible, our evolutionary framework offers critical advantages for dynamic ride-hailing systems where traffic conditions, passenger requests, and driver positions change continuously. Below we clarify why fixed-prompt approaches are insufficient and explain the need to use the proposed evolutionary framework:\n- **Dynamic adaptability to system state changes**: Fixed prompts or fine-tuned models without feedback loop generate static outputs that cannot evolve in response to real-time changes in the ride-hailing system. Our framework uses feedback-driven prompt evolution via Harmony Search to refine the LLM-generated objective functions iteratively, ensuring alignment with the current state of the ride-hailing system. \n- **Data/compute efficiency**: Fine-tuning requires extensive data, computational resources, and retraining when domain shifts occur, e.g., new cities, different demand patterns. Our method is much quicker and more efficient as we only need to evolve the prompt, guided by the downstream optimization feedback. \n- **Mitigation of partial observability**: Fine-tuned model with fixed prompts lack full foresight of the environment dynamics, leading to misalignment. By using LLM as a meta-objective designer within a closed-loop evolutionary process, we overcome the partial observability challenge and reduce reliance on human-crafted heuristics.\n\nTherefore, while fine-tuning small LLMs can yield good performance in static or narrowly scoped settings, it is insufficient to address the dynamic and real-time decision-making challenges inherent in ride-hailing dispatch systems. In contrast, our proposed framework, embedding LLMs within an evolutionary optimization loop, enables **real-time adaptivity**, **training-free generalization**, and **solver-grounded solution quality**, which are essential for practical deployment in large-scale urban mobility systems.\n\n**Comment 3**. Would it be impossible to design effective objective functions using rule-based methods? Is the use of LLMs necessary for generating high-quality objectives? How complex are the behaviors demonstrated by LLMs in the experiments?\n\n**Response**: Thanks for the reviewer's comments. While rule-based objectives are feasible, our experiments reveal that traditional designs often rely on fixed linear/quadratic penalties or manually tuned weights. Therefore, it is difficult to adapt dynamically to complex systems like ride-hailing, where passenger requests, traffic, and taxi availability vary rapidly over time. Also, rule-based methods often rely on extensive domain expertise and can only be modified under manual recalibration. \n\nAn example of a generated objective from our experimental results is provided below:\n`````\ndef dynamic_obj_func(self):\n      # Improved objective function to enhance taxi utilization by considering arrival times, dynamic weights, and better load  balancing.\n        print(\"Creating dynamic objectives for Assignment Model\")\n        \n        # Cost component 1: Passenger waiting time from request to taxi arrival\n        cost1 = gb.quicksum(\n            self.y[v,p] * max(self.taxi[v].arrival_time - self.passenger[p].arrTime, 0)\n            for v in self.taxi.keys() for p in self.passenger.keys())\n        \n        # Cost component 2: Taxi detour time from current position to passenger pickup\n        cost2 = gb.quicksum(\n            self.y[v,p] * self.distMatrix[self.taxi[v].start_pos][self.passenger[p].origin]\n            for v in self.taxi.keys() for p in self.passenger.keys())\n        \n        # Cost component 3: Passenger trip time from pickup to drop-off\n        cost3 = gb.quicksum(\n            self.y[v,p] * self.distMatrix[self.passenger[p].origin][self.passenger[p].destination]\n            for v in self.taxi.keys() for p in self.passenger.keys())\n        \n        # Cost component 4: Penalty for multiple assignments per taxi\n        cost4 = gb.quicksum(\n            (gb.quicksum(self.y[v,p] for p in self.passenger.keys())) ** 2\n            for v in self.taxi.keys())\n        \n        # Calculate dynamic weights based on idle taxis\n        idle_taxis = sum(\n            1 for v in self.taxi.values()\n            if v.arrival_time <= self.passenger[next(iter(self.passenger.keys()))].arrTime)\n        \n        weight_factor = 1 + (idle_taxis / len(self.taxi) if len(self.taxi) else 0)\n        \n        # Cost component 5: Load balancing penalty across taxis\n        load_balance_factor = 1.5 * weight_factor\n        cost5 = gb.quicksum(\n            (gb.quicksum(self.y[v,p] for p in self.passenger.keys()) -\n             (len(self.passenger) / len(self.taxi))) ** 2 * load_balance_factor\n            for v in self.taxi.keys())\n        \n        # Adjust weights for better taxi utilization\n        weights = [\n            weight_factor * 2.5,   # Higher weight on passenger waiting time\n            weight_factor * 1.75,  # Consider taxi detour time\n            weight_factor * 1.25,  # Ensure efficient passenger trips\n            1200 * weight_factor,  # Strong penalty for overloading taxis\n            300 * weight_factor    # Enhanced load balancing importance\n        ]\n        \n        # Combine cost components\n        costs = [cost1, cost2, cost3, cost4, cost5]\n        \n        # Set objective\n        objective = sum(w * c for w, c in zip(weights, costs))\n        self.model.setObjective(objective, gb.GRB.MINIMIZE)\n`````\n\nClearly, the LLM-generated objective (as shown above) has multiple cost components with dynamic weights that adjust based on system states, which is beyond typical rule-based design. It effectively captures trade-offs among passenger wait time, detour time, and load balancing, adapting in real time to system conditions. Our experiments (see Tables 2 and 5 in the paper) validate its effectiveness: In the Manhattan downtown scenario, especially large cases, our method achieves delay costs that are approximately **∼40% lower** than the best-performing rule-based method, aligning with the final drop-off time trends in Figures 7 and 8 in the paper Appendix."}, {"review_id": "d8LVrd09br", "reviewer": "Reviewer_8CYJ", "summary": "This paper proposes a hybrid method that integrates LLMs with exact solvers to address dynamic sequential decision-making in mobility-on-demand systems (which has an extremely high-dimensional search space and thus necessitates decomposition into high- and low-level optimization problems). The LLM in the method acts as a meta-optimizer that dynamically formulates high-level objectives via prompt-based evolution (employing the HS algorithm for the evolutionary process). The exact solver then addresses the optimization problems established at the corresponding time step. The method decomposes the ride-hailing problem hierarchically into assignment and routing subproblems, with LLMs bridging the cognitive gap between high and low levels by generating semantically-rich objectives that implicitly encode low-level routing dynamics.", "questions": "NA", "limitations": "NA", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "This paper addresses a practical problem, which I consider highly valuable. On the one hand, it proposes a novel problem formulation for ride-hailing systems and provides a formal analysis of the associated search space complexity. On the other hand, this work is the first to integrate LLMs with exact solvers for sequential dynamic decision-making systems. The proposed method is training-free and overcomes the data inefficiency challenges inherent in reinforcement learning, as well as the rigidity of manually designed objectives in traditional decomposed optimization. A key feature is its ability to dynamically adjust the high-level optimization problem according to $\\mathcal{S}\\_t$ and $H\\_{t-1}$. The experimental results are also promising.\n\nMy concerns are as follows:\n- What is the rationale for using the HS algorithm? Can it be replaced by other EAs?\n- What are the specific objects designed by FunSearch and EoH in the experiments? Can the HS algorithm be substituted with FunSearch or EoH? I think that these two methods essentially also design functions.\n- The ablation study on the operators of the HS algorithm is missing.\n- Due to the use of LLMs, the worst-case runtime of the proposed method is difficult to predict.\n- I think the author(s) should clarify that all information contained in $\\mathcal{S}\\_t$ and $H\\_{t-1}$ positively contributes to the method's performance. Besides, could author(s) provide a specific example of $\\mathcal{S}\\_t$ and $H\\_{t-1}$?\n- Is it possible to dynamically adjust the parameters $\\alpha, \\beta, \\gamma$ to improve the performance of manual objectives? Moreover, the experiments do not specify the settings of these parameters.\n- Can multiple high-level objectives be pre-designed using LLMs and the appropriate objective be automatically selected at each time step? I think this could significantly reduce both runtime and the cost associated with using LLMs.", "was_revised": true, "final_justification": "Thank you for your response. Most of my concerns have now been satisfactorily addressed, so I have increased my rating. I appreciate the effort involved in preparing additional results during the rebuttal period. However, I strongly recommend that the author(s) include a more comprehensive and robust empirical study in the camera-ready version (if accepted), with additional experiments to demonstrate how the new method improves performance. I believe these results will interest a broad audience.", "rebuttal": "**We appreciate the reviewers’ thorough feedback. We have grouped the identified weaknesses and concerns into the following comments for clarity and response.**\n\n**Comment 1**. What is the rationale for using the HS algorithm? Can it be replaced by other EAs?\n\n**Response**: Thanks for the reviewer's comments. We select HS because it can address the key challenge for evolving LLM prompts: encoding complex problem context, such as environment states and past objectives, into natural language in a clear and effective way. Traditional evolutionary strategies like Genetic Algorithms (GA), which rely on **crossover** and **mutation**, often produce semantically inconsistent text, **disrupting the alignment between prompt structure and evaluation scores**, making fitness unreliable. Furthermore, they **require complex parsing to maintain coherence**, ultimately reducing robustness and effectiveness in prompt optimization. HS avoids these issues by using two mechanisms: **Memory Consideration (HMCR)**: Copies full parent prompts, preserving structure and associated fitness. **Pitch Adjustment (PAR)**: Applies minimal edits, enabling controlled refinement without breaking grammar or semantics.\n\n**Comment 2**. What are the specific objects designed by FunSearch and EoH? Can HS be substituted with FunSearch or EoH? I think that these two methods essentially also design functions.\n\n**Response**: Thanks for the reviewer's comments. While FunSearch and EoH indeed involve function design, their formulation and target applications differ fundamentally from ours in several key aspects:\n- **Problem Nature**: FunSearch and EoH target *static, single-agent* combinatorial problems such as TSP and bin packing, where solutions can be constructed greedily without temporal coupling. In contrast, our task involves *multi-agent, dynamic decision-making* under time-dependent constraints (see Eqs. (1g)–(1m) in the paper). \n- **Output Type**: FunSearch/EoH prompt LLMs to directly generate *executable code function*, e.g., `def next_location(unvisited, current_pos)`. Our framework instead prompts the LLM to generate *mathematical objective functions*, which are optimized by a mixed-integer linear programming (MILP) solver to produce globally feasible, coordinated assignments. \n- **Conformity to System Constraints**: FunSearch/EoH bypass optimization solvers and cannot enforce hard constraints. More strict prompt design is required to avoid generating infeasible solutions in multi-constraint settings. However, FunSearch and EoH cannot guarantee to generate feasible solutions when the problem constraints are complex. On the other hand, HS in our method evolves high-level objective structures that remain compatible with MILP formulations (see paper Appendix A.9.2–A.9.3 for Gurobi-specific prompt constraints). This integration of semantic generation (LLM) and numerical grounding (solver) is central to our approach. \n\nThus, FunSearch and EoH cannot be used to substitute HS. Doing so will lead to a lack of system-wide interactions, as FunSearch and EoH mostly generate local, next-visit decisions without global coordination. Furthermore, there will be more infeasible solutions, as essential hard system constraints cannot be enforced.\n\n**Comment 3**. The ablation study on the operators of the HS algorithm is missing.   \n \n**Response**: Thanks for the reviewer's comments. Due to the main paper’s strict page limit, the HS parameter study was included in Appendix A.1.1 under the title 'Evolutionary hyperparameter study' (line 604). We can relocate this section to the main body of the paper accordingly. In this Appendix subsection, we describe how the parameters HMCR and PAR are adjusted to control the selection frequency of each operator.\n\n**Comment 4**. Due to the use of LLMs, the worst-case runtime of the proposed method is difficult to predict.\n\n**Response**: Thanks for the reviewer's comments. We would like to clarify that this paper focuses on the **conceptual and algorithmic design of a novel LLM-guided optimization framework**. In practical deployments, especially for latency-sensitive applications like transportation, LLMs can be deployed locally to minimize network delays and ensure tighter runtime guarantees. \n```\nTo understand our problem, we also provide the run time in Tables I and II below for our experiments. Although runtime grows with simulation scale, it remains significantly below the actual simulation duration. For example, Case 7 in Table I (20min simulation) completes in ~4 minutes, despite multiple optimization rounds, confirming the framework's practical runtime efficiency.\n```\nTable I. Run time (s) for Manhattan cases \n| Sim Duration (s) | 600 | 600 | 600 | 900 | 900 | 900 | 1200 | 1200 | 1200 |\n|--------|--------|--------|--------|--------|--------|--------|--------|--------|--------|\n| Cases | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 | Case 9 |\n| Run Time (s) | 127.27| 138.89 | 107.33 | 178.89 | 194.51 | 178.43 | 233.50 | 208.70 | 196.33 |\n\nTable II. Run time (s) for Chicago cases \n| Sim Duration (s) | 600 | 600 | 600 | 900 | 900 | 900 | 1200 | 1200 | 1200 |\n|--------|--------|--------|--------|--------|--------|--------|--------|--------|--------|\n| Cases | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 | Case 9 |\n| Run Time (s) | 119.31 | 133.19 | 116.82 | 141.21 | 149.80 | 178.70 | 223.89 | 155.63 | 216.92 |\n\n**Comment 5**. I think the author(s) should clarify that all information contained in $S_t$ and $H_{t-1}$ positively contributes to the method's performance. Besides, could author(s) provide a specific example of $S_t$ and $H_{t-1}$?\n\n**Response**: Thanks for the reviewer's comments. We have clarified and expanded this explanation in the main body of the paper.  They are all basic inputs to LLM, and eliminating them will make it difficult for LLM to understand the current environmental traffic situation.\n- Current system state at time $t$: $S_t$ = {$V_t, P_t, C_t$}, where $V_t$ is the set of active vehicles, each with location, availability status and estimated arrival time. $P_t $ denotes the set of pending passenger requests, including pickup and drop-off locations and request times. $C_t $ is environment constraints, such travel time dynamics. \n- Historical Decision Trajectory up to $t-1$: $H_{t-1}$ = {$(a_0, f_0), (a_1, f_1), ..., (a_{t-1}, f_{t-1})$}, where $a_k$ is the LLM-generated high-level objective for time $k$ (not the dispatch/routing action obtained the optimization solver) based on $S_k$. $f_k$ is a scalar fitness score (average passenger delay) used to evaluate the effectiveness of $a_k$ via the downstream solver.\n\nBoth components are critical in constructing the LLM prompt. $S_t$ provides essential real-time context, while $H_{t-1}$ enables the LLM to identify temporal patterns, such as repeated failures or regional imbalances. These form the dynamic prompt component ${P}_{dyn}$, which enables the LLM to adaptively refine the objective function based on evolving conditions. \nThe following JSON illustrates the structure at step $t$ for dynamic prompt:\n`````\n{\"taxi info\": [ \"Taxi 0: start_pos=24, est_arr_time=0s\", ... \"Taxi 59: start_pos=8, est_arr_time=100s\"], \"passenger info\": [ \"Passenger 1: origin=8, dest=32, requestT=181s\", ... \"Passenger 31: origin=5, dest=32, requestT=248s\"],\n \"llm response\": {\"obj description\": \"New objective function focuses on minimizing passenger waiting time, taxi travel time, and balancing assignments with adjusted weights.\", \"obj code\": \"def dynamic_obj_func(self):\\n  cost1 = ...\\n  cost2 = ...\\n  cost3 = ...\\n  weights = [10, 20, 1000]\\n  objective = sum(w*c for w,c in zip(weights, [cost1, cost2, cost3]))\\n  self.model.setObjective(objective, gb.GRB.MINIMIZE)\" }, \"response format\": \"Your obj function is correct. Gurobi accepts your obj.\", \"evaluation score\": 320}\n`````\n**Comment 6**. Is it possible to dynamically adjust the parameters $\\alpha$, $\\beta$, $\\gamma$ to improve the performance of manual objectives? Moreover, the experiments do not specify the settings of these parameters.\n\n**Response**: Thanks for the reviewer's comments.  In our experiments, the weights of manual objective methods (as baseline methods at Tables 2 and 5 in the paper) strictly follow fixed values from prior work [2][4] (cited in the paper), ensuring consistency and fair benchmarking. \n\nThe objective cost is defined as: $$obj = \\sum_{i} weight_{i} \\times cost_{i}$$ \nAdjusting weights ($\\alpha$, $\\beta$, $\\gamma$) is just one degree of freedom, also, modifying only the weights of the fixed cost does not yield much improvement. However, LLM can generate flexible weight and cost adjustments.\n\nAppendix 9.4 (line 959) in the paper provides several examples of high-level objectives generated by our LLM-guided method. As captured in Figures 17 and 18 in the paper, the model combines creative costs with novel dynamic and non-linear penalties that are challenging to define manually.\n\n**Comment 7**. Can multiple high-level objectives be pre-designed using LLMs and the appropriate objective be automatically selected at each time step? I think this could significantly reduce both runtime and the cost associated with using LLMs.\n\n**Response**: Thanks for the reviewer's suggestions. Pre-generating a library of high-level objectives can reduce LLM inference overhead, but in dynamic environments like real-time mobility systems, static objectives may quickly become suboptimal due to changing conditions (e.g., demand spikes or congestion). To maintain adaptability while ensuring low latency and cost efficiency, LLMs can be locally deployed and queried selectively for just-in-time updates. Furthermore, recent LLM advancements (e.g., quantized models and lightweight adapters) make it practical to update or fine-tune objective-generation logic periodically based on recent performance trends, achieving both responsiveness and efficiency."}]

Author rebuttal:

Meta-review: This paper proposes a hierarchical optimization framework, and four reviewers have submitted their recommendations for the manuscript. This paper it the first closed-loop integration of LLM with mathematical optimization for sequential, dynamic decision-making. Although everyone has pointed out some merits, they also have raised some concerns, such as the motivation for HS algorithms, scalability and its application to real-world scenarios. During the rebuttal period, the authors’ feedback has helped on clarifying the reviewers’ concerns. Thus, the average score 4 is both above the average levels.
Decision: Accept (poster)