Trip-Based Sampling Optimization

• The paper presents a novel optimization framework that leverages timetabled bus trips and trip chains to maximize spatial-temporal sensor coverage in urban settings.
• It employs a sequential three-stage formulation—including bus-line pre-selection, minimum-fleet sizing, and sensor allocation—to efficiently manage computational complexity while ensuring high coverage.
• The joint bi-level formulation co-optimizes scheduling and sensor placement, reducing sensor requirements by up to 22% and substantially increasing grid–time coverage.

Trip-based sampling is an optimization framework for the deployment of a limited number of mobile sensors on fleet buses, aiming to maximize spatial-temporal coverage for drive-by sensing tasks (such as air quality, traffic state, and road roughness monitoring). The methodology explicitly incorporates timetabled bus trips, exploits the structure of trip chains (ordered sequences of trips served by the same bus), and reconciles operational constraints on minimal fleet size with coverage maximization, all while maintaining computational tractability at city scale (Ji et al., 2023).

1. Problem Setting and Definitions

The spatial domain is discretized into grids $g \in G$ (e.g., $1\,{\rm km} \times 1\,{\rm km}$), and time is segmented into intervals $t \in T$ of fixed length $\Delta$ (e.g., 60 min). The bus network consists of $|\mathcal{L}|$ lines, each with a fixed timetable. A trip $i\in I_l$ on line $l$ is specified as $(p^i,q^i,t_{li},\tau_i)$: start/end terminals, scheduled departure time, and duration. Dead-heading time $t_{lij}$ defines the non-service interval between consecutive trips $i,j$ by the same vehicle.

A trip chain $c$ is a feasible, time-respecting sequence of timetabled trips a physical bus can serve in a day. Chains are feasible only if the dead-heading constraints are satisfied: $t_{li}+\tau_i+t_{lij} \leq t_{lj}$ for all transitions in the chain. Up to $N_S$ identical sensors may be installed, each assigned to a trip chain (i.e., bus), conferring sensing ability for the whole chain.

Coverage is defined at the grid–time pair $(g,t)$ level: $n_{gt}=1$ if at least one sensor-equipped bus is in grid $g$ during $t$, zero otherwise. Spatial ($w_g$) and temporal ($\mu_t$) weights (normalized to sum to 1) model heterogeneous monitoring priorities. The global coverage (sensing reward) is

$$\Phi = \sum_{g\in G} w_g \sum_{t\in T} \mu_t n_{gt}.$$

Operational constraints include complete fulfillment of the timetable with the minimal fleet size (minimum-fleet principle), ensuring sensor assignment does not compromise service.

2. Sequential Three-Stage Formulation

Stage A: Bus-Line Pre-Selection

To reduce problem size, a set cover is solved to select a minimal subset $L\subset\mathcal{L}$ of lines covering at least a fraction $\gamma$ of all reachable grids (with $\gamma=1$ yielding full coverage). Let $\delta_{gl}=1$ if line $l$ covers grid $g$. The binary program minimizes $\sum_{l}x_l$ subject to constraints ensuring sufficient grid coverage and logical consistency.

Stage B: Minimum-Fleet Sizing per Line

For each selected line $l\in L$, a bipartite matching is solved to minimize the number of buses required while chaining trips into feasible sequences. Variables $y_{lij}$ indicate whether trip $j$ is served immediately after $i$. The minimum fleet for line $l$ is $N_l^{\rm min}=N_{I_l}-\max_{y}\sum_{i,j}y_{lij}$, with $N_{I_l}$ the number of trips on $l$. Matched pairs are extracted to form all trip chains $C=\cup_l C_l$.

Stage C: Sensor Allocation to Trip Chains

Sensor assignment is phrased as a 0-1 integer program over all trip chains. Binary variable $z_c$ flags instrumented chains. For each trip and grid–time pair, indicator $n_{igt}$ marks if trip $i$ covers $(g,t)$. Constraints ensure no more than $N_S$ sensors are assigned, and that every covered grid–time pair is supported by at least one equipped bus.

These distinct stages—pre-selection, fleet sizing, sensor allocation—frame the trip-based sampling approach as a sequence of linked optimization problems.

3. Joint Bi-level Formulation

The joint bi-level model addresses the sub-optimality arising from fixing trip chains in advance, instead co-optimizing scheduling and sensing assignments per line.

• Upper Level: Across all lines, integer variables $m_l$ distribute the available $N_S$ sensors, maximizing total coverage by blending information on how many sensors to assign per line (subject to per-line saturation $K_l$).
• Lower Level (per line): For a given $m_l$, the problem is to select $m_l$ trip chains to be instrumented, optimizing the coverage contributed by that line. Variables $(z_c,\xi_{cij},\xi_{ci},n_{gt})$ model which chains and trip transitions are chosen, and their resulting sensing impact.

The bi-level structure is separable by line, allowing parallel solution, with sensor allocation at the upper level guided by lower-level computations of attainable coverage for each $m_l$.

The two levels interact only through the mappings $m_l \leftrightarrow q_{gt}^{(l, m_l)}$, with $q_{gt}^{(l, m)}$ denoting grid–time coverage from line $l$ equipped with $m$ sensors.

4. Algorithmic Workflow and Computational Properties

The algorithm proceeds as follows:

1. Line Pre-Selection: The set cover step significantly reduces the problem size, selecting $\ll |\mathcal{L}|$ relevant lines.
2. Per-Line Optimization: For each chosen line,
   • The fleet sizing (bipartite matching) is solved in $O(|I_l|^{2.5})$ time (max-flow/assignment).
   • Model reduction prunes superfluous link variables $\xi_{cij}$ where idle times exceed a threshold $\delta$, preserving optimal fleet size and saving up to 90% in problem dimensionality.
   • For $m=0,1,2,\dots$, the pruned mixed-integer program is solved to find $\Phi_l(m)$ and associated coverage. Computation stops when further sensors do not increase coverage (at saturation $K_l$).
3. Global Sensor Allocation: The upper-level knapsack-like integer program (in $|L|$ variables) allocates $N_S$ sensors to lines.

Each line's lower-level problem is independent, and the reduced $|L|$ after pre-selection enables sub-linear scaling in $|\mathcal{L}|$. In contrast, a naïve vehicle-based approach is combinatorial in the total number of buses or trip chains.

5. Empirical Study: Chengdu Case

A comprehensive real-world test covers $400\,{\rm km}^2$ within Chengdu’s 4th Ring Road, with $400$ one-kilometer grids and service from $7\,$am to $10\,$pm. Three temporal granularities ($\Delta=60,\,90,\,120$ min) are examined, $\mu_t=1/|T|$. Spatial weights $w_g$ are derived from traffic and emission data.

Of $167$ bus lines, pre-selection (with $\gamma=1$) yields $L=38$ lines ensuring full grid coverage. These require a minimum fleet of $684$ buses for $6,006$ trips. To achieve $90\%$ coverage of grid–time pairs at $\Delta=60$ min, the sequential approach requires $49$ sensors; the joint bi-level model needs only $38$ (a reduction of $22\%$). The number of grids fully covered in every interval increases by $41$–$238\%$ under the joint model. Almost every line saturates at $K_l \leq 2$ sensors for $\Delta=60$ min, and $K_l = 1$ for $\Delta \geq 90$ min.

Computation times are significantly improved after pre-selection: $1,313\,s$ for fleet-sizing on all $167$ lines versus $90\,s$ on $38$ lines; sensor allocation MILPs take minutes instead of $12\,h$. Pruning with $\delta \approx 100$ min (idle time) reduces solution time by $25$–$60\%$ without degrading coverage.

Aspect	Sequential Approach	Joint Bi-level Approach
Sensors for 90% cover	49	38
Increase in 100% grids	Baseline	+41–238%
Saturation per line	$K_l\leq2$ (60 min)	$K_l\leq2$ (60 min)

6. Model Extensions and Practical Recommendations

Multiple model extensions are available for operational realism:

• Service gaps: Dummy trips $D_l$ with fixed time windows (e.g., for breaks or charging) can be inserted, with chain assignment constraints.
• Bus relocations: Forbidden by taking $t_{lij}=\infty$ or penalized with a multi-objective cost term ($\beta t_{lij} \xi_{cij}$).
• Operational costs: Additional terms for total fleet size ($\alpha N_l^{\min}$) and dead-heading ($\sum \beta t_{lij} y_{lij}$ or $\sum \beta t_{lij} \xi_{cij}$).
• Uncertain service times/speeds: Addressable through robust or stochastic variants, or corrected via subsequent data processing.

A practical rule of thumb is to assign one sensor per selected line and prioritize a second sensor to lines with large one-way trip durations, to close temporal coverage gaps for coarse $\Delta$.

The trip-based methodology thus tightly integrates the combinatorics of fleet scheduling with the needs of optimal spatial-temporal sensor allocation. It achieves near-optimal city-scale coverage under realistic operational constraints and computational budgets, with the decoupling by lines ensuring both tractability and deployment feasibility (Ji et al., 2023).