Pareto Front Particle Swarm Optimizer for Discrete Time-Cost Trade-Off Problem Saman Aminbakhsh 1 and Rifat Sonmez 2 Abstract: Intensive heuristic and metaheuristic research efforts have focused on the Pareto front optimization of discrete time-cost trade-off problem (DTCTP). However, very little success has been achieved in solving the problem for medium and large-scale projects. This paper presents a new particle swarm optimization method to achieve an advancement in the Pareto front optimization of medium and large-scale construction projects. The proposed Pareto front particle swarm optimizer (PFPSO) is based on a multiobjective optimization environment with novel particle representation, initialization, and position-updating principles that are specifically designed for simultaneous time-cost optimization of large-scale projects. PFPSO brings several benefits for the discrete time-cost optimization, such as an adequate representation of the discrete search space, fast convergence properties, and improved Pareto front optimization capabilities. The computational experiment results reveal that the new particle swarm optimization method outperforms the state-of-the-art methods, both in terms of the number of Pareto front solutions and computation time, especially for medium and large-scale problems. A large number of nondominated solutions are achieved within seconds for the first time, for a problem including 720 activities. The proposed Pareto front particle swarm optimizer provides a fast and effective method for optimal scheduling of construction projects. DOI: 10.1061/(ASCE)CP.1943-5487.0000606. © 2016 American Society of Civil Engineers. Author keywords: Scheduling; Costs; Optimization; Algorithms; Multiple objective analysis; Project management. Introduction In the critical path method (CPM) scheduling, the durations of ac- tivities can be shortened by allocating additional manpower and machinery resources. Expediting critical path activities enables reducing indirect costs and avoiding potential delay penalties, but at the cost of increased direct costs. This trade-off between time and cost is known as the time-cost trade-off problem. The objective of general time-cost trade-off problem is to identify the set of time-cost modes (alternatives) that will minimize the total cost of the project. The importance of time-cost trade-off problem has been recog- nized since the development of the CPM (De et al. 1995). Early research on the time-cost trade-off problem assumed the relation between time and cost to be continuous (Kelley and Walker 1959; Fulkerson 1961; Siemens 1971; Goyal 1975). In recent years there has been increased attention toward the discrete version of the problem because of its practical relevance. In the literature, two types of the discrete time-cost trade-off problem (DTCTP) have been commonly studied. The objective of the first problem is to determine the optimal set of time-cost modes that will minimize the total cost, including direct and indirect costs, penalties, and bo- nuses for a given project deadline. The second problem is a multi- objective optimization problem and involves determination of the complete and nondominated time-cost profile over the set of feasible project durations (Vanhoucke and Debels 2007), which is called the Pareto front. A few metaheuristic approaches have been presented for the single objective, first type of DTCTP (Hegazy 1999; Elbeltagi et al. 2007; Vanhoucke and Debels 2007; Abdel-Raheem and Khalafallah 2011; Sonmez and Bettemir 2012). Numerous studies have focused on achieving the Pareto front for the DTCTP. Genetic algorithms (Feng et al. 1997; Zheng et al. 2005; Kandil and El-Rayes 2006; Eshtehardian et al. 2009), ant colony optimization (Ng and Zhang 2008; Afshar et al. 2009; Zhang and Ng 2012), harmony search (Geem 2010), and particle swarm optimization (PSO) (Yang 2007; Zhang and Li 2010) are among the metaheur- istic solution procedures proposed for the Pareto front optimization of DTCTP. Zahraie and Tavakolan (2009), Ashuri and Tavakolan (2012), and Ashuri and Tavakolan (2015) considered the Pareto front optimization of resources along with the time and cost. Despite a large amount of the research has focused on the Pareto front optimization for the DTCTP, the majority used problems in- cluding up to 18 activities in computational experiments. Very few of the existing methods can be applied to Pareto front optimi- zation of real-life construction projects that typically include more than 300 activities (Liberatore et al. 2001). Besides, a few methods that are tested for large-scale problems require a signifi- cant amount of computation time because of the inherent difficulty in solving the problem. The parallel genetic algorithm (GA) of Kandil and El-Rayes (2006) required 136.5 h on a single processor, and 15.4 h over a supercomputing cluster of 50 processors to achieve a large number of Pareto front solutions for a problem in- cluding 720 activities. The main objective of this paper is to fill the gap in the multi- objective DTCTP literature by developing a particle swarm optimi- zation method that can achieve successful Pareto front solutions in a short amount of computation time for large-scale problems. The proposed method is based on the novel principles for representa- tion, initialization, and position-updating of the particles that are specifically designed for the multiobjective DTCTP. 1 Ph.D. Candidate, Dept. of Civil Engineering, Middle East Technical Univ., Ankara 06531, Turkey. E-mail: saman.aminbakhsh@metu.edu.tr 2 Associate Professor, Dept. of Civil Engineering, Middle East Technical Univ., Ankara 06531, Turkey (corresponding author). E-mail: rsonmez@ metu.edu.tr Note. This manuscript was submitted on October 13, 2015; approved on April 19, 2016; published online on June 20, 2016. Discussion period open until November 20, 2016; separate discussions must be submitted for in- dividual papers. This paper is part of the Journal of Computing in Civil Engineering, © ASCE, ISSN 0887-3801. © ASCE 04016040-1 J. Comput. Civ. Eng. J. Comput. Civ. Eng., 04016040 Downloaded from ascelibrary.org by Middle East Technical University on 08/24/16. Copyright ASCE. For personal use only; all rights reserved.