Document Type : Research Paper

Authors

• Sara Bagherzadeh Rahmani ¹
• Javad Rezaeian ²
• Ahmad Ebrahimi ³

¹ Ph.D. Student in Industrial Management, Department of Industrial Management, Faculty of Management and Economics,, Science and Research Branch, Islamic Azad University, Tehran, Iran

² Associate Professor, Department of Industrial Engineering, Faculty of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran

³ Assistant Professor, Department of Industrial Management and Technology, Faculty of Management and Economics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Abstract

In today’s project-based organizations, where multiple projects are executed concurrently within work teams, human resources play a crucial role in the success or failure of these organizations. Consequently, human resources are recognized as one of the most essential resources for these organizations, and their optimization can significantly increase productivity while reducing organizational time and costs. This underscores the importance of effective human resource management and highlights the need for special attention to this area. Therefore, this study presents a mixed-integer nonlinear programming model for the multi-objective project scheduling problem with resource constraints, multi-skilled personnel allocation and the assignment of projects to work teams. The mathematical model of this research includes the multiple objectives of simultaneous minimization of the total costs of setting up work teams and the use of human resources and the total flow time of projects. To make the model more realistic, the effect of learning is also considered. Subsequently, a diverse set of test problems at varying scales was designed. Then, the Multi-Objective Artificial Immune System (MOAIS) algorithm and the Non-dominated Sorting Genetic Algorithm II (NSGA-II) were utilized to solve the problems. The results demonstrate the superior performance of the NSGA-II algorithm compared to the MOAIS algorithm.
Introduction
Human resource management is one of the fundamental pillars of organizational success and, alongside financial and technological resources, plays a crucial role in process optimization and achieving strategic objectives. Optimal workforce allocation and effective project scheduling enhance productivity, reduce costs, and ensure the efficient utilization of resources. Teamwork and knowledge sharing facilitate learning and skill development, which, in complex projects with limited resources, lead to shorter project completion times and improved organizational efficiency.
Accordingly, this study addresses the project scheduling problem by considering human resource constraints, multi-skilled work teams, setup times, varying project start times, and work-team dependent learning effect. The primary objective is to simultaneously minimize the total costs of setting up work teams and the use of human resources, as well as the total project flow time. To achieve this, a Mixed-Integer Linear Programming (MILP) model is developed.
Given that the problem is NP-hard, employing metaheuristic algorithms is essential for obtaining near-optimal solutions within a reasonable computational time. This research utilizes two metaheuristic algorithms, NSGA-II and MOAIS.
The findings of this study provide valuable insights for project managers and decision-makers, aiding in optimized project scheduling, efficient workforce allocation, and enhanced organizational productivity.
Research Background
In this section, we mention only a few of the most relevant studies to the current one. Su et al. (2021) explored team formation in project scheduling and presented a simple mathematical model for task scheduling in single-skilled workgroups with restricted access to resources, aiming to minimize makespan (C_max). In their model, workers were assigned to fixed groups, and tasks were allocated based on processing time and workforce availability. They proposed a hybrid genetic algorithm with a bin-packing strategy to solve the problem.
 Mozhdehi et al. (2024) developed a mixed-integer mathematical model for multi-project scheduling with limited resources and multi-skilled workforce. They considered workforce agility, which improves either through collaborative teamwork and knowledge-sharing with more skilled colleagues or by dedicating more time to skill development. Their results indicated that incorporating workforce agility into project scheduling models significantly reduces project completion time.
Methods
In this study, a Mixed-Integer Linear Programming (MILP) model was developed to address the multi-project scheduling problem with multi-skilled work teams. The model integrates human resource constraints, setup times, and work-team dependent learning effect, ensuring a practical and efficient scheduling framework.
For solving the model, the single-objective version was first handled using the Branch and Bound algorithm in Lingo software. Then, for the multi-objective version, two metaheuristic algorithms, NSGA-II and MOAIS, were implemented to generate high-quality trade-off solutions.
To assess and compare the performance of these algorithms, a set of test problems of different scales (small, medium, and large) was designed and solved. The Taguchi Experimental Design Method was employed to fine-tune the key algorithm parameters, optimizing efficiency and accuracy.
Evaluating the performance of multi-objective metaheuristic algorithms is more complex than that of single-objective optimization due to the presence of non-dominated solutions that cannot be strictly ranked. In this study, the following key metrics were used to assess solution quality and diversity:

Number of Pareto Solutions (NPS)
Mean Ideal Distance (MID)
Diversity Metric (DM)
Spread of Non-dominated Solutions (SNS)

Discussion and Results
Results of sensitivity analysis reveals that increasing the learning rate of work teams significantly reduces project completion time. This finding underscores the importance of incorporating learning effects in multi-skilled workforce scheduling models. With a higher learning rate, teams execute tasks more efficiently and in less time, directly contributing to organizational productivity improvements.
Furthermore, computational results indicate that in small to medium-sized problems, there is no significant performance difference between NSGA-II and MOAIS. However, in large-scale problems, NSGA-II outperforms MOAIS. This superiority is attributed to NSGA-II’s population evolution mechanism, which enables a broader exploration of the solution space and prevents premature convergence to local optima. In contrast, MOAIS, due to its elitist nature, primarily focuses on replicating high-quality solutions, avoiding exploration in other regions of the search space. This increases the likelihood of getting trapped in local optima, thereby reducing search diversity. Furthermore, performance comparison results indicate that NSGA-II demonstrates superior Pareto front coverage and convergence to optimal solutions compared to MOAIS.
Conclusion
This study investigated the project scheduling problem considering human resource constraints, multi-skilled work teams, setup times, varying project start times, and work-team dependent learning effects. The primary objective was the simultaneous minimization of the total costs of setting up work teams and the use of human resources and the total flowtime of projects. To achieve this, a Mixed-Integer Linear Programming (MILP) model was developed, and its performance was evaluated through sensitivity analyses and numerical experiments. The results demonstrated that the proposed model performed effectively under various constraints and exhibited high accuracy and efficiency.
Given the NP-hard nature and multi-objective characteristics of the problem, two metaheuristic algorithms, NSGA-II and MOAIS, were implemented to solve it. The algorithm parameters were fine-tuned using the Taguchi Experimental Design Method, and their performance was compared across different problem sizes. Computational results indicated that while both algorithms performed similarly in small to medium-sized problems, NSGA-II outperformed MOAIS in large-scale instances. Further analysis revealed that MOAIS, due to its elitist-based nature, primarily focuses on replicating high-quality solutions, often avoiding broader exploration within the solution space. This characteristic increases the likelihood of getting trapped in local optima, reducing solution diversity. In contrast, NSGA-II, through its non-dominated sorting mechanism, allows lower-fitness solutions to participate in the evolution process, leading to broader solution space exploration and preventing premature convergence.
For future research, it is recommended to extend the model by incorporating additional operational assumptions, such as activity failure probabilities, simultaneous consideration of learning and forgetting effects, rework processes, and uncertainty of certain parameters in the problem. Furthermore, exploring more advanced heuristic and hybrid metaheuristic algorithms is suggested to enhance the efficiency of the solution approach.

Keywords

• Project Scheduling
• Multi-Skilled Working Teams
• Work-Team Dependent Learning Effect
• NSGA-II
• MOAIS

Main Subjects

• Industrial management