A Study on Construction Schedule Management Methods under Disruptive Conditions Based on Combinatorial Thinking

To support the development of the proposed combinatorial sensitivity framework, this section defines the core terms and parameters used throughout the methodology.

Disturbance refers to any unexpected event that disrupts task execution during construction, such as delays, resource shortages, or schedule interruptions. In this study, disturbances are modelled as proportional extensions of task durations to reflect time-based uncertainty.

Disturbing Task Combination denotes a set of construction tasks that are simultaneously subjected to disturbance within a simulation scenario. Each combination represents all tasks affected under a given disturbance event, with the number of tasks in the set determined by the perturbation parameter.

Multi-task Perturbation describes a disturbance scenario in which multiple tasks within a selected combination are subjected to specified changes in duration. These changes can be uniform or task-specific, depending on the simulation design or the nature of disturbances observed in actual engineering projects.

Perturbation Parameter defines the number of tasks selected from the entire set of construction activities in the CPM-based network. Each combination represents one simulation unit in the sensitivity analysis.

These conceptual definitions ensure consistent terminology and provide the contextual background for the mathematical formulations presented in the following sections.

To efficiently represent the logical dependencies among construction tasks, this study formalizes the Critical Path Method (CPM) network structure into a matrix-based model. Let the project consist of n construction tasks; two core matrices are constructed as follows:

The core of representing a construction schedule network via matrices lies in capturing the dependencies between activities (Maheswari & Varghese, 2005). This is commonly achieved using a task dependency matrix, denoted as D. Given n tasks, D is an n × n matrix where D_ij indicates whether task i depends on task j. If task i must start only after task j is completed, then D_ij= 1; otherwise, D_ij = 0 (Uma Maheswari et al., 2006).

The matrix is represented as follows: (1) D=D11⋯D1n⋮⋱⋮Dn1⋯Dnndij=1,if task i depends on the completion of j0,otherwise \matrix{{{\rm{D}} = \left[ {\matrix{{{{\rm{D}}_{11}}} \hfill & \cdots \hfill & {{{\rm{D}}_{1{\rm{n}}}}} \hfill \cr \vdots \hfill & \ddots \hfill & \vdots \hfill \cr {{{\rm{D}}_{{\rm{n}}1}}} \hfill & \cdots \hfill & {{{\rm{D}}_{{\rm{nn}}}}} \hfill \cr}} \right]} & {{{\rm{d}}_{{\rm{ij}}}} = \left\{{\matrix{{1,} \hfill & {{\rm{if}}\,{\rm{task}}\,{\rm{i}}\,{\rm{depends}}\,{\rm{on}}\,{\rm{the}}\,{\rm{completion}}\,{\rm{of}}\,{\rm{j}}} \hfill \cr {0,} \hfill & {{\rm{otherwise}}} \hfill \cr}} \right.} \cr}

In construction projects, task durations determine execution length and influence inter-task dependencies as well as the overall project duration. The task duration matrix T is typically an n × 1 vector, where n is the total number of tasks in the project. Each element Ti in the matrix denotes the duration of task I (Ma et al., 2019).

The matrix is represented as: (2) T=[T1,T2,…,Tn]T, T = {[{T_1},{T_2}, \ldots,{T_n}]^T},

Let t_i denote the start time of task i, t_j the start time of task j, and d_j the duration of task j. Based on the above, the following scheduling constraint must be satisfied: (3) ti≥tj+dj if Dij=1, {t_i} \ge {t_j} + {d_j}\,\,if\,\,{D_{ij}} = 1,

In construction schedule management, the cost constraint matrix C is used to represent the cost incurred by each task when consuming resources over specific time periods. Let n denote the total number of tasks, m the total number of resource types, and T the number of time periods. During task and resource allocation, the cost matrix C is an n × m matrix, where each element C_ij represents the unit cost incurred by task i for utilizing resource j.

The matrix is represented as: (4) C=C11⋯C1n⋮⋱⋮Cm1⋯Cmn C = \left[ {\matrix{{{C_{11}}} \hfill & \cdots \hfill & {{C_{1n}}} \hfill \cr \vdots \hfill & \ddots \hfill & \vdots \hfill \cr {{C_{m1}}} \hfill & \cdots \hfill & {{C_{mn}}} \hfill \cr}} \right

Assuming that task i is allocated resource j during time t, the incurred cost is calculated as: (5) Costijt=Aij×Cij {Cost}_{ijt} = {A_{ij}} \times {C_{ij}}

To enable automatic identification and dynamic adjustment of the critical path, this study integrates the D, T and C matrices with an improved Floyd algorithm to accumulate path weights and trace the longest path, thereby accommodating path reconfiguration under disturbances (Ding et al., 2023). After the structured modelling of the task network is completed, the next step is to identify critical tasks and quantify their impact on the project timeline.

Based on combinatorial thinking, this study models multiple tasks as an integrated disturbance unit and develops a dual-index sensitivity framework—covering schedule and cost—to identify task clusters with significant influence on project objectives (Hammad et al., 2020). To quantitatively evaluate the impact of task-level disturbances on project objectives, two categories of sensitivity indicators are proposed in this study:

The Schedule Sensitivity Index (SSI) quantifies the extent to which a multi-task combination disturbance impacts the overall project duration. (6) Schedule Sensitivity Index(SSIk)=ΔTkT∑i∈kΔTiTi×100% {\rm{Schedule}}\,{\rm{Sensitivity}}\,{\rm{Index}}({\rm{SSI}}_k) = {{{{\Delta {T_{\rm{k}}}} \over {\rm{T}}}} \over {\sum\nolimits_{i \in k} {{{\Delta {T_i}} \over {{T_i}}}}}} \times 100\%

Where:

• ΔTₖ - the amount of change in the total duration of the project caused by the disturbance combination k [days],

• T - original project duration before disturbance [days],

• ΔT_i - applied disturbance for task i [days],

• T_i - original duration of task i [days],

• k - task combination.

The Cost Sensitivity Index (CSI) reflects the joint impact of schedule disturbances on project cost and duration across multiple task combinations: (7) Cost Sensitivity Index(CSIk)=ΔCkCΔTk/∑i∈Ckwi⋅(Ti) {\rm{Cost}}\,{\rm{Sensitivity}}\,{\rm{Index}}({\rm{CSI}}_k) = {{{{\Delta {C_{\rm{k}}}} \over C}} \over {\Delta {T_k}/\sum\nolimits_{i \in {C_k}} {{w_i} \cdot ({T_i})}}}

Where:

• ΔC_k - total cost increment caused by disturbing task combination k [CNY],

• C - original total project cost [CNY],

• w_i - resource weighting factor of task i, defined as: (8) wi=TiCiΣj∈kTjCj, {w_i} = {{{T_i}{C_i}} \over {{\Sigma_{j \in k}}{T_j}{C_j}}}, Where:

  • C_i - unit cost per time of task i,

  • ∑_j∈Ck T_jC_j - the total resource cost of all tasks within the disturbed task combination k,

  • w_i - represents the relative resource weight of task i, determined by its time-cost contribution within the combination.

To ensure consistent interpretation of the Schedule Sensitivity Index (SSI) and Cost Sensitivity Index (CSI), both indices are categorized into three levels-High, Medium, and Low-using the same classification method.

Specifically, a percentile-based segmentation approach is applied: all simulation results are sorted in ascending order for each index, and the 33.33^rd percentile (P33) and 66.67^th percentile (P66) are selected as threshold points to divide the results into three segments. To reduce the uncertainty in classification near the percentile threshold values, after obtaining the results, the original P33 and P66 boundary points were slightly adjusted according to the distribution density of adjacent data points. This smoothing processing method enhances the interpretability while retaining the classification framework based on percentile values.

This data-driven method allows the classification scheme to remain consistent in logic while flexibly adapting to the distributional characteristics of each index. The exact numerical ranges for each sensitivity level are presented in Section 3 based on the actual value distributions observed in the case study.

To further reveal the sensitivity of different resource types to project cost changes under multi-task disturbances, this study decomposes the cost increment of each task into subcomponents for each resource type and introduces corresponding resource weights. The Cost Sensitivity Index is thus formulated as follows: (9) CSIK(λ)=1C∑i∈k(∑j=1mλj⋅ΔCij) {CSI}_K(\lambda) = {1 \over C}\sum\limits_{i \in k} {(\sum\limits_{j = 1}^m {{\lambda_j} \cdot \Delta {C_{ij}}})} where:

• m - the total number of resource types,

• ΔC_ij - the cost increment of task i for resource type j after the disturbance,

• λ_j - the weight assigned to resource type j (which can be determined based on its importance or volatility at different project stages).

By integrating schedule and cost sensitivity indices, this study introduces a CPM-based management model enhanced by combinatorial scanning. A parameterized task grouping algorithm is used to apply structured perturbations and evaluate dual-index responses across task combinations. The resulting sensitivity values serve as the basis for prioritization in schedule planning. The specific process is shown in the Figure 2.

Figure 2:

Flowchart of schedule control using combinatorial scanning and dual-index sensitivity evaluation

The disturbance levels adopted in this study were primarily determined based on two considerations: (1) In real construction environments, external disruptions such as weather delays or resource constraints often affect multiple tasks simultaneously and with comparable impact; and (2) applying uniform disturbances across task combinations allows for comparative evaluation of their relative schedule and cost sensitivities, thus helping to identify structurally vulnerable task groups.

As shown in Table 1, the improved model based on combinatorial thinking surpasses the traditional CPM in task analysis granularity, responsiveness to disturbances, and resource optimization.

The case project involves a bridge engineering, from which the primary construction workflow was extracted to generate a construction task network diagram (Figure 3) and a detailed task schedule (Table 2). Based on construction logic, the task dependency matrix D and the task duration vector T were established. An improved Floyd algorithm was then applied to identify the critical path.

Figure 3:

Construction double code arrow diagram

The results indicate that the critical path of the project is A→B→L→M, with an original total duration of 210 days; tasks on the critical path play a decisive role in schedule control.

A parameterized scenario simulation approach was adopted, and based on Pareto analysis, N = 4 tasks were selected to cover key nodes on the critical path. A complete enumeration method was employed to generate all C134=715 {\boldsymbol{C}}_{{\boldsymbol{13}}}^{\boldsymbol{4}} = 715 possible task combinations. Perturbations of 10% and 20% were applied to each combination, and the total project duration was recalculated to derive the corresponding Schedule Sensitivity Index (SSI). Table 5 presents the simulation results for a selection of representative task combinations.

These perturbation levels and uniform resource weights are adopted for simulation clarity and comparability; future studies will incorporate heterogeneous task-specific disturbances, stochastic duration/cost variations, and empirically calibrated resource weights.

Under resource constraints, schedule control often requires trade-offs with cost allocation. This study introduces the Cost Sensitivity Index (CSI) within a combinatorial thinking framework to quantify the joint schedule-cost impact caused by multi-task combination disturbances. For the case of N = 3, C = 1, 000, 000 a full enumeration approach was used to generate all C133=286 {\boldsymbol{C}}_{{\boldsymbol{13}}}^{\boldsymbol{3}} = 286 task combinations. Schedule-cost perturbations were applied to each task combination, and the variations in bi-objective outcomes were analysed. The results are presented in Table 6.

Based on the classification results of the Schedule Sensitivity Index (SSI) and Cost Sensitivity Index (CSI) in sections 3.2 and 3.3, the following scheduling recommendations are proposed for different management priorities under the simulated disturbance scenarios. These recommendations are derived from controlled simulation conditions with uniform disturbance levels and resource weights and should be interpreted within the scope of these assumptions:

• (1)

  For schedule-driven scenarios: Priority should be given to task combinations with high Schedule Sensitivity Index (SSI), such as ADLM and BIJK. These combinations have the greatest potential to delay project completion under disruptions and thus require proactive management and resource protection. Recommended actions include allocating schedule buffers, securing critical resources in advance, and monitoring progress milestones for early warnings.

• (2)

  For cost-constrained scenarios: Combinations with high-Cost Sensitivity Index (CSI)—such as AGF and EGF— should be closely monitored and managed, even if they do not significantly affect the overall schedule. Early intervention is essential to avoid disproportionate cost escalation.

• (3)

  For combinations with both high SSI and high CSI—such as ADG and DHI— adopt integrated strategies that combine schedule protection with cost control, such as phased resource allocation and contingency planning. Combinations with low SSI and CSI values (e.g., AEF, EHM) can be used for flexible resource reallocation to support higher-risk tasks without compromising key objectives.