A Markov-Based Algorithm for Determining the Optimal Project Due Date in PERT Networks considering Non-Renewable Resources

Document Type : Original Article

Authors

1 M.Sc. Student., School of Industrial Engineering, Iran University Science & Technology, Tehran, Iran.

2 Associate Prof., School of Industrial Engineering, Iran University Science & Technology, Tehran, Iran.

3 MSc., School of Industrial Engineering, Iran University Science & Technology, Tehran, Iran.

Abstract

Introduction and Objectives. Project management under uncertainty, particularly when facing discontinuous resource constraints, represents one of the fundamental challenges in project planning. In real project environments, fluctuations in activity durations and resource shortages frequently lead to deviations from predetermined schedules and increased operational costs. These challenges are more evident in time-sensitive projects such as construction projects, new product development, and research initiatives. The primary objective of this research is to develop an analytical framework for assessing the probability of timely project completion across different time intervals and utilizing this assessment to determine the most favorable project termination time with maximum economic benefit. This study seeks to address the fundamental question of how to establish project delivery deadlines that minimize delay costs while enabling early utilization, despite uncertainties in activity durations and discrete resource limitations.
Methods. The current research employs an enhanced critical path network modeling approach. In this model, activity durations are defined as probabilistic variables with specified distributions. The resources required for activity execution are considered non-renewable and are allocated to the project at predetermined intervals. For analyzing the temporal behavior of projects, discrete-time stochastic processes have been utilized. This approach enables the examination of various project states at distinct time points and calculates the probability of project completion at each time interval. Subsequently, an algorithm has been developed to identify all possible project progress states and compute the project completion probability function at each time point. Additionally, a combined cost-benefit optimization function has been developed that considers the probability of project completion at different times, calculating both delay-induced costs and benefits from early completion. The optimal project termination time is determined through analysis of this function, aiming to maximize net profit from on-time or early project delivery.
Findings. Results from implementing the model on various project samples indicate that resource allocation scheduling significantly impacts the probability of timely project completion. Analysis of the project completion probability function revealed that in certain resource distribution scenarios, the probability of on-time completion increases substantially, leading to reduced delay costs and increased benefits from early delivery. Case studies demonstrated that in many instances, selecting a delivery time slightly different from the median completion probability time can result in significant financial performance improvement. For example, in one examined project, choosing a delivery time just 5% earlier than the median time led to a 15% increase in net project profit. Comparison of different resource allocation scenarios showed that the proposed approach can serve as an effective decision-making tool at various managerial levels. The research also established that combining project completion probability analysis with economic optimization provides a comprehensive solution for determining optimal project delivery times.
Conclusion. The model presented in this research offers a novel approach for analyzing projects under conditions of resource and activity duration uncertainty. By employing stochastic processes and the proposed algorithm, the model enables determination of optimal project termination times based on economic profit maximization. Findings indicate that this framework can serve as an effective tool for project managers in establishing realistic and economically sound delivery deadlines. Future research should focus on extending this model by incorporating qualitative factors influencing project management.

Keywords

Main Subjects

References

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Journal of Industrial Management Perspective
Volume 15, Issue 4 - Serial Number 60
December 2025
Pages 253-279

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