## 4 Model Development
### DP Procedure
The process then builds on the solutions derived from the initial subproblems to address subsequent phases, thus ensuring that the cumulative strategy is not only effective but also efficient. For the study in question, the implementation strategy adopted to structure the problem was based on a comprehensive exploration of the DP landscape. Specifically, all possible transitions in each state of the problem were examined. This exhaustive approach ensured that all possible combinations of tasks and activities that could be performed within the given time frame were explored. The aim was to identify the particular sequence or combination that produced the most cost-effective outcome. This rigorous approach to DP ensures that the final strategy is both cost-optimal and feasible given the constraints. For readers seeking a detailed breakdown of the algorithm used, Table 3 explains each step in detail. The pseudocode of the algorithm used in the proposed model in this study is listed in Algorithm 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Step** & **Name** & **Content** \\ \hline \multirow{4}{*}{1} & & \(\bullet\) The first step involves the definition of task states. A state encapsulates the progress and status of the construction at a particular point in time. Two primary components characterize a state: \\  & state & - Completed tasks: This refers to tasks that have been performed up to the current state. It provides a snapshot of progress and an understanding of the tasks that have been completed. \\  & definition & - Remaining time: An essential metric that indicates the expected time remaining for a specific task or the entire project. This component supports efficient resource allocation and effective scheduling. \\  & & \(\bullet\) Once states have been defined, it is essential to define the rules that govern state transitions. \\  & transition & Transitions depend on three key aspects: \\  & rules & - Choice: Often there may be several paths available for a transition. The choice of the optimal path may be influenced by constraints such as cost or available resources. \\  & & - Dependency: There are tasks whose onset depends on the completion of previous tasks. Such dependencies must be carefully mapped. \\  & & - Concurrency: Identifying tasks that can be performed concurrently can speed up the overall process. It is important to identify and exploit such concurrency opportunities. \\  & & \(\bullet\) Inherent in recursive algorithms, including DP, is the need for a termination condition, known as the base case. In the field of construction optimization: \\
3 & base case definition & - The base case could manifest itself when all tasks are completed or when the expected remaining time for a particular task becomes zero. It is essential to define this base case precisely to ensure the correct termination of the algorithm. \\  & formulation & \(\bullet\) The culmination of the problem formulation process is the formulation of the objective function. This function quantitatively represents the cost associated with a particular state. The primary objective, especially in a construction paradigm, is to minimize this cost. This cost could be the sum of financial expenditures, time consumption, and resource utilization. An appropriately formulated objective function is central to the optimization process and provides a clear optimization objective for the DP algorithm. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dynamic programming (DP) implementation approach.