Finally, Sections " Conclusion" and " Recommendations for future work" conclude the study and outline future research directions. Section " Computational experiments" presents the results from the validation and application of our model. Section " Model development" outlines the development of our hybrid multi-verse optimizer model. Section " Literature review" summarizes the literature on the time–cost trade-off problems. The rest of this paper is organized as follows. This model also significantly enhances the decision-making ability of decision-makers. Therefore, a hybrid multi-verse optimizer model (hDMVO) was developed by combining the MVO and the SCA to provide efficient solutions for medium- and large-scale DTCTPs and other optimization problems that can be applied in actual construction projects. Despite the availability of several existing methods, they are not fully equipped to solve large-scale DTCTPs. The resolution of large-scale DTCTPs is a crucial aspect in the management of any construction project. Therefore, hDMVO will achieve a reasonable balance between the exploration and the exploitation phases, which ensures that the algorithm can achieve global optimization and become an appropriate metaheuristic method for solving the DTCTP. Concurrently, good search area exploitation by the algorithm is guaranteed by SCA through the fact that the value closest to the global optimum is stored in a variable as the target and is never lost during the optimization. The hDMVO algorithm was developed by preserving MVO's mechanisms of white and black holes to ensure good exploration of the search area by MVO. Two algorithms with opposite advantages and disadvantages motivated us to develop a hybrid algorithm between MVO and SCA for optimal exploration and exploitation of the search area based on the strengths of each algorithm to achieve a balance between the two mechanisms. Specifically, its search area exploitation mechanism is not clearly expressed therefore, it easily encounters fast convergence 6, which results in local optimization. SCA has been utilized to address optimization challenges in diverse domains since 2016 5. In addition, the study shows that SCA converges significantly faster than PSO, GA, ACO, etc. The results of the test problems show that SCA can explore different regions of the search space, avoid local optimization, converge towards global optimization, and effectively exploit the promising region of the search space during optimization. The Sine Cosine Algorithm (SCA) 4 was developed for focusing on the exploration and exploitation of the search space during optimization. However, MVO has issues in balancing the exploration and exploitation mechanism of the search area and limitations in the search area exploitation during fast convergence, thus resulting in local optimization 3. The results show that the MVO algorithm can provide competitive, even superior results than those of other algorithms in most tested optimization problems. For result assessment, MVO is compared with other metaheuristic algorithms, such as particle swarm optimization (PSO), Genetic Algorithm (GA), Ant colony optimization (ACO), etc. Mirjalili, Mirjalili 2 proposed a multi-verse optimizer (MVO) algorithm inspired by the Big Bang theory to satisfy the need for solving single-and multi-objective optimization problems. The Pareto front is a multi-objective optimization problem to simultaneously optimize both project cost and time 1. Meanwhile, the time optimization problem is aimed at choosing alternative solutions to shorten the project implementation time while ensuring that the project cost does not exceed the revenue on the early operation of the project. The objective of the cost optimization problem is to minimize the total cost under specific conditions, including project implementation time and penalty costs for delays. Cost optimization, time optimization, and Pareto front are three common forms of time–cost trade-off problems. The importance of optimization in a construction project has been emphasized for decades as it is used to find the ideal plan and schedule for completing a project. The productivity of different components of a project can be increased by optimization. In project management, optimization is a highly useful tool to satisfy desired objectives under specific constraints.
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