Enhancing Urban Air Mobility Scheduling Through Declarative Reasoning and Stakeholder Modeling
Abstract
1. Introduction
2. Related Work
2.1. UAM Scheduling: Challenges and Domain-Specific Approaches
2.2. Resource-Constrained Scheduling: MILP and Heuristics
2.2.1. RCPSP and MILP Formulations
2.2.2. Heuristic and Metaheuristic Approaches
2.3. Declarative Approaches: Answer Set Programming (ASP) in Scheduling
2.4. Stakeholder-Centric Scheduling
3. Preliminaries
3.1. Scenario I: Vertiport Operations as an RCPSP
- Full Turnaround Sequence: Arrival → Landing → Passenger Disembarkation → Battery Charging → Passenger Embarkation → Pre-flight Checks → Departure. This is a sequence of the general operations that a vehicle drops off passengers, proceeds to required maintenance (like charging), boards new passengers, and then departs. Resources like landing pads, gates or stands, and charging stations are utilized sequentially.
- Quick Drop-off Sequence: Arrival → Landing → Passenger Disembarkation → (Minimal Checks or Preparation) → Departure. This might be used for repositioning flights or short hops where extensive ground services like charging are not immediately required.
- Cargo Operations Sequence: Arrival → Landing → Cargo Unloading → Cargo Loading → Pre-flight Checks → Departure. This is for transporting cargo. Instead of passenger-related jobs, it handles cargo-related jobs.
- Charging-Focused Sequence: Arrival → Landing → Battery Charging → Pre-flight Checks → Departure. This sequence is only for eVTOL to visit the vertiport for charging without handling passengers or cargo.
3.2. RCPSP
- J: a set of jobs ,
- R: a set of resources,
- S: a list of the order in which the tasks in the jobs will be executed,
- T represents the planning horizon, which is a list of potential completion time steps for jobs,
- : the time required to complete job j,
- The formula represents the required unit of resource r to complete job j, and
- : the capacity of resource r.
- task(A) signifies that A is a task to be scheduled.
- step(I) denotes that I is a discrete time step within the planning horizon.
- The predicate start(A,I) means task A is assigned to begin at time step I.
- dur(A,D) indicates that task A has a processing duration of D time steps.
- Resource requirements are specified by rsreq(A,R,AM), meaning task A requires AM units of resource R.
- The predicate assigned_res(TS,A,R,AM1) is used to track that AM1 units of resource R are assigned to task A at time step TS.
- sum_assigned_res(TS,R,AM) is to calculate the total amount AM of resource R to be assigned at time step TS. This includes all active tasks at this time step.
- limit(R,L) specifies that resource R has a maximum capacity of L units.
- psrel(P,S) is a precedence relation. It means that task P must complete before task S starts.
- Finally, etask(A) identifies A as an end task (specifically, the tail dummy task in this context) whose start time is part of the objective function.
- Equation (8) ensures that every task A is assigned exactly one start time I from the available time steps. The construct 1 {…} 1 is an ASP choice rule, guaranteeing one successful assignment.
- Equation (10) calculates the total resource usage. For a given time step TS and resource R, the component of the rule sums (#sum) all amounts AM1 of that resource assigned to any task A (via assigned_res(TS,A,R,AM1)). The result is unified with AM.
- Equation (12) is the objective function. It instructs the ASP solver to find a schedule that minimizes the value I, where I is the start time of the tail dummy task (identified by estart(I), which is derived from start(A,I), etask(A)). Hence, minimizing the start time of this final dummy task effectively minimizes the overall project makespan.
3.3. Multiple Schedules
3.4. Requirements
3.4.1. [Req1] Requirement 1: Punctuality
3.4.2. [Req2] Requirement 2: Boarding of Additional Seats
3.4.3. [Req3] Requirement 3: Managing Delivery Delays
4. Problem Formulation
4.1. Problem Statement
- Input: Instances and requirements
- Output: A set of start times and a solution of requirements
- Process: Calculate a schedule for under the consideration of where .
4.2. Solution Overview
5. Proposed Solution
5.1. Listings of Requirements
5.1.1. Solution for Req1
5.1.2. Solution for Req2
5.1.3. Basic Solution for Req3
5.2. Combining Listings with Procedural Codes
Algorithm 1: RCPSP-REQ-ASP |
Algorithm 2: RCPSP-REQ-ASP-ITER |
6. Experiments and Discussions
6.1. Experiments
6.1.1. Toy Example
6.1.2. Numerical Experiments
6.2. Discussions
6.2.1. Performance Issue
6.2.2. What We Have Not Covered Here
- LP Relaxation: Our approach to RCPSP is focused on MILP, not linear programming. This is because our MILP formulation in Section 3.2 was to clearly introduce discrete and countable resource management through integer programming. We can narrow down the possible result sets for a specific RCPSP case. This can be performed by heuristically performing LP relaxation. We can also establish lower bounds [38,76,77,78,79]. However, we chose MILP to obtain an exact solution. Choosing MILP let us easily transition to the ASP formulation, and this allowed for a simpler representation of the requirements.
- Graph-Based Algorithmic Approach: This scheduling problem differs from a simple pathfinding problem in that it does not involve navigating a physical environment. Decisions made in one job plan by the RCPSP will impact other job plans. This approach is quite similar to cooperative pathfinding [49]. Since this perspective is relatively new in research, we need more scenarios and simulation results. This will help us apply this problem and approach in practice.
6.2.3. Future Directions
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
ASP | Answer set programming |
ATC | Air Traffic Control |
CDM | Collaborative decision making |
DE | Differential evolution |
eVTOL | electric vertical takeoff and landing |
GA | Genetic algorithm |
MILP | Mixed-integer linear programming |
MPF-JSS | Multi-resource partial-order flexible job-shop scheduling |
PSO | Particle swarm optimization |
RCPSP | Resource-constrained project scheduling problem |
UAM | Urban air mobility |
UATM | UAM Air Traffic Management |
CDCL | Conflict-driven clause learning |
SAT | boolean SATisfiability problem |
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Job | Duration | Resources Required | Successors | ||
---|---|---|---|---|---|
Resource 0 | Resource 1 | Resource 2 | |||
1 | 5 | 3 | |||
2 | 2 | 1 | 1 | ||
3 | 1 | 2 | 2 | ||
4 | 5 | 3 | 1 | 2 | |
5 | 2 | 3 | 1 | 3 | |
6 | 4 | 1 | |||
7 | 2 | 3 | |||
8 | 5 | 1 | 3 | ||
9 | 5 | 1 | 6, 7 | ||
10 | 4 | 1 | 9 |
Function | MILP | ASP | Description |
---|---|---|---|
Objective | Equation (2) | Equation (12) | It minimizes the start time of the tail dummy. |
Constraints | Equations (3) and (6) | Equation (7) | Each task should be assigned only once. |
Equation (4) | Equations (9)–(11) | The required resources for each task should be allocated for the duration from its beginning, and the sum of all allocated resources at every time step should not exceed the limit. | |
Equation (5) | Equation (8) | Every pair of tasks in the precedence relationship should follow the order. |
Set 1 | Set 2 | |
---|---|---|
Aux. facts for | The set is {“tx1”, “tx2”, “tx3”}. owner(“req1”, “tx3”), owner(“req2”, “tx1”), owner(“req3”, “tx2”), tree(“tx3”, 9..10), tree(“tx1”, 1..5), tree(“tx2”, 6..8). | |
Aux. facts for | deadline(“tx3”, 25), preferable_range(“tx1”, 1, 2), text_extension(0, “tx2”, 5), delay_range(0..5). | deadline(“tx3”, 15), preferable_range(“tx1”, 2, 3), text_extension(0, “tx2”, 4), delay_range(0..4). |
Additional facts | delay_margin(0, 0). | |
punctual_candidate(9, 20), deadline_met(“tx3”). | punctual_candidate(9, 10), deadline_met(“tx3”). | |
excessed(4, 1), excessed(5, 1), excess_in_total(0, 2). | excessed(4, 1), excessed(5, 1), excess_in_total(0, 2). | |
actual_delay(0, 4). | actual_delay(0, 1). | |
Total Duration | estart(25). | estart(19). |
MILP | ASP | |||||
---|---|---|---|---|---|---|
Time [s] | Step [#] | Succ. [#] | Time [s] | Step [#] | Succ. [#] | |
(Min|Median|Max) | (Min|Median|Max) | (Min|Median|Max) | (Min|Median|Max) | |||
10 | 00.108|000.256|041.252 | 09|16.0|25 | 100 | 0.000|000.031|000.064 | 09|16.0|25 | 100 |
20 | 00.646|015.134|455.195 | 13|21.0|32 | 94 | 0.080|000.271|002.156 | 13|20.5|32 | 100 |
30 | 27.312|132.038|424.036 | 20|23.5|30 | 20 | 0.459|001.887|375.537 | 18|26.0|42 | 98 |
40 | — | — | 0 | 2.516|026.192|572.800 | 20|31.0|38 | 71 |
50 | — | — | 0 | 7.943|148.850|554.881 | 28|35.0|42 | 24 |
Time [s] | Time [s] | Step [#] | Punctuality [#] | |
---|---|---|---|---|
(Min|Median|Max) | (Min|Median|Max) | (Min|Median|Max) | ||
Plain | 0.095|0.446|30.191 | — | 13|21|40 | 35 |
Req | 0.136|0.657|04.082 | 01.795|03.378|08.275 | 16|25|45 | 1000 |
Iter | 0.121|0.648|04.285 | 06.669|13.864|35.994 | 16|25|45 | 1000 |
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Kim, J.; Kim, K. Enhancing Urban Air Mobility Scheduling Through Declarative Reasoning and Stakeholder Modeling. Aerospace 2025, 12, 605. https://doi.org/10.3390/aerospace12070605
Kim J, Kim K. Enhancing Urban Air Mobility Scheduling Through Declarative Reasoning and Stakeholder Modeling. Aerospace. 2025; 12(7):605. https://doi.org/10.3390/aerospace12070605
Chicago/Turabian StyleKim, Jeongseok, and Kangjin Kim. 2025. "Enhancing Urban Air Mobility Scheduling Through Declarative Reasoning and Stakeholder Modeling" Aerospace 12, no. 7: 605. https://doi.org/10.3390/aerospace12070605
APA StyleKim, J., & Kim, K. (2025). Enhancing Urban Air Mobility Scheduling Through Declarative Reasoning and Stakeholder Modeling. Aerospace, 12(7), 605. https://doi.org/10.3390/aerospace12070605