## 4 A Review of Variants in Project Scheduling Problem
### Based on Solution Methods  
In real life, project constraints present only one of the challenges in project planning. Over the past two decades, various techniques of planning project scheduling under resource constraints were proposed, implemented, and controlled, which can broadly be categorized into exact and approximate methods. In fact, it is known that the Resource-Constrained Project-Scheduling Problem has a history longer than 40 years. There are two approaches, exact and approximate, for solving the problem. Each of these approaches has disadvantages and advantages. Exact approaches for solving the RCPSP were first proposed by [PERSON] [88], followed by [PERSON] and [PERSON] [89]. Exact methods provide optimal results. Exact methods include Dynamic Programming, Constraint Programming, and Mixed Integer Programming, including Lagrangian Relaxation, Zero-One Programming, and Branch and Bound (B&B). These algorithms only solve small problems and cannot find optimal solutions in reasonable computation times. Thus, these algorithms cannot work  
Figure 4: Percentage of research by classification of cost-based objective function studied. Figure 3: Percentage of research by classification of time-based objective function studied. for large and complex projects. Many research papers have employed exact methods to deal with RCPSP [90; 91]. To overcome the computational burden of exact methods, at the expense of optimality, approximate methods are an alternative. These methods, instead of searching the entire problem space, search only a part of it, so the results may not be optimal. They aim, instead, to achieve an approximate solution, but they can solve huge problems quickly. The approaches are categorized into two different groups: (1) approaches based on heuristic methods such as PR-based approaches, Forward and Backward Improvement (FBI), and Network analysis; and (2) approaches based on meta-heuristic methods such as genetic algorithm (GA), Tabu Search (TS), Simulated Annealing (SA), and Ant Colony Optimization (ACO) [14; 16; 30; 47; 67; 92; 93]. Recent years have witnessed a rise in hybrid optimization approaches for solving complex variants, like the Multi-Mode Resource-Constrained Project Scheduling Problem (MRCSP). For instance, [PERSON] et al. [40] proposed an integer programming-based approach for multi-mode scheduling, while [PERSON] et al. [77] explored a bi-objective hybrid model combining MILP with GAs. Other hybrid algorithms merge TS with decomposition methods [94] or integrate local searching within evolutionary algorithms to improve convergence speed and solution diversity. In addition, Artificial Intelligence (AI) and machine learning techniques are increasingly applied in project scheduling. [PERSON] et al. [54] proposed a genetic programming framework for stochastic RCMPSP under project insertions. Neural networks have also been explored for predictive modeling of activity durations and decision support in dynamic environments (e.g., [PERSON] et al. [80] and [PERSON] et al. [95]). These AI-based approaches enhance adaptability and robustness in uncertain and multi-mode settings. An overview of the project scheduling problem can be seen in Figure 5.  
Tables 5 and 6 show the works that used each exact or approximate algorithm, respectively, to solve the problem. Figure 5: An overview of project-scheduling problems and solving methods. Initially, RCMPSP variants were solved mainly using mathematical models and PR-based heuristics. With increasing complexity and the emergence of new solution techniques, a broader array of proposed solution methodologies was expanded. Figure 6 shows the percentages of use of approximate and exact algorithms. It is clear that meta-heuristics are the most widely used (65%). To identify the primary solution methods employed for addressing the problem in recent years, Figure 7 illustrates the research conducted since 2013, focusing on methods that have been examined in a minimum of 12 studies.