Abstract
Purpose
This research aims to propose self-adaptive ant colony optimization (SACO) with changing parameters for solving time-cost optimization (TCO) problems to assist the relevant construction management firm with their technological tool.
Design/methodology/approach
A SACO with changing parameters based on information entropy has been employed to model TCO problem, which overcomes the intrinsic weakness of premature convergence of the basic ant colony optimization by adjusting parameters according to mean information entropy of the ant system. A computer simulation with Matlab 7.0 based on a prototype example has been carried out on the basis of SACO for TCO problem.
Findings
The test results show that the SACO for TCO model can generate a better cost under the same duration and achieve a better Pareto front than other models. Therefore, the SACO can be regarded as a useful approach for solving construction project TCO problems.
Research limitations/implications
Further research on selection parameters should be conducted to further improve the robustness of the SACO for TCO model.
Practical implications
The modelling results can help the construction management to good result of TCO problems in construction sites.
Originality/value
A new approach to study the TCO model is proposed based on SACO.
Keywords
Citation
Li, H. and Li, P. (2013), "Self-adaptive ant colony optimization for construction time-cost optimization", Kybernetes, Vol. 42 No. 8, pp. 1181-1194. https://doi.org/10.1108/K-03-2013-0063
Publisher
:Emerald Group Publishing Limited
Copyright © 2013, Emerald Group Publishing Limited
1 Introduction
Time and cost are the two important objectives of construction projects, and they are intricately related. The total cost for each project is the sum of the direct and indirect cost. Direct cost commonly represents labor, materials, equipment, etc. But indirect cost generally represents overhead cost such as supervision, administration, consultants, and interests. Direct cost grows at an increasing rate as the project time is reduced from its original planned time. However, indirect cost continues for the life of the project and any reduction in project time means a reduction in indirect cost. Therefore, there is a trade-off between the time and cost for completing construction activities.
The time-cost optimization (TCO) problem is a multiobjective problem, which attempts to strike a balance between resource allocation costs and project schedule duration. This has led to the development of the TCO concepts. The biggest challenge of TCO problems is on analyzing the time-cost implication of all possible combinations within a short period of time and at a reasonable cost given the huge amount of construction activities (Ng et al., 2000).
There is a need to construct models and algorithms for general construction project scheduling optimization. These models should be able to account for real-world uncertainties, as well as correlation between cost and duration of numerous scheduling options available. In general, the required models should account for the stochastic nature of the scheduling problem, and be applicable to actual size problems. Construction schedule trade-off problems have been tackled repeatedly in the literature using different methods.
Techniques and algorithms for modeling time-cost problems can be classified into three categories, namely the heuristic methods (Fondahl, 1961; Siemens, 1971; Moselhi, 1993), mathematical approaches (Kelly, 1961; Robinson, 1975; Elmaghraby, 1993; Burns et al., 1996), and evolutionary-based optimization algorithms (EOAs) (Elbeltagi et al., 2005). Detailed discussions on the heuristic methods and mathematical approaches can be found in Zheng et al. (2005). Although the two methods have their specific advantages, neither heuristic methods nor mathematical approaches can solve the multiobjective TCO problem efficiently.
Many researchers have explored more advanced approaches to obtain global optimal solutions for TCO problems, such as EOAs. Genetic algorithms (GAs) are based on the mechanics of natural selection and genetics to search through the decision space for the identification of optimal solutions, which have been commonly used for deriving optimal solution for multiobjective problem domains. Feng et al. (1997) developed a GA model that was essentially an improvement of a hybrid model devised by Liu et al. (1995) and Feng et al. (2000) further developed a stochastic GA model for construction time-cost trade-off problems. Zheng (2004) proposed using the GA approach for solving the multiobjective TCO problems. The evolution process of GAs predicted the survival and characteristic of offspring on the basis of unveiling the characteristic of their parents (Ahmed and Eldin, 2004). Despite its benefit, the processing time of GAs model to generate a reasonable solution can be excessive. Except GA, other EOAs techniques as inspired by different natural processes including the ant colony optimization (ACO) (Dorigo and Gambardella, 1996), mimetic algorithms (Moscato, 1989), particle swarm optimization (Kennedy and Eberhart, 1995), shuffled frog leaping approach (Eusuff and Lansey, 2003), etc. were employed by Elbeltagi et al. (2005) for solving discrete time-cost trade-off problems. The results of Elbeltagi's study confirmed that ACO was among the best for handling time-cost trade-off problems both in term of solution quality and processing time. So, Ng and Zhang (2008) and Zhang and Ng (2012) had employed ant colony approach to solve the TCO problem, and the results showed that the ant colony system approach was able to generate better solutions without utilizing much computational resources.
But there are some intrinsic weaknesses of ACO models. The first problem is its tendency of premature convergence. As the number of iterations increases, the results tend to converge toward a set of values. On the other hand, selection of parameters could affect the performance of the ACO. An ACO model is sensitive to the parameters selected, as they can affect the convergence speed and the quality of solutions. Some researchers have developed some approaches to improve the ACO model. Wei (2010) used the path selection controlled by information entropy and random perturbations strategy to realize adaptive regulation. In order to solve the premature convergence problem of the basic ACO algorithm, a promising modification based on the information entropy is proposed. The main idea is to evaluate stability of the current space of represented solutions using information entropy, which is then applied to tuning of the algorithm's parameters. The path selection and evolutional strategy are controlled by the information entropy self-adaptively. Simulation study and performance comparison with other ACO algorithms and other meta-heuristics on the travelling salesman problem (TSP) show that the improved algorithm, with high efficiency and robustness, appears self-adaptive and can converge at the global optimum with a high probability (Li and Li, 2007).
In this study, a self-adaptive ant colony optimization (SACO) approach is employed to derive a set of best solutions for TCO problems, combining with the modified adaptive weight approach (MAWA). To beginning with, fundamental concepts of time-cost problem and basic ACO are presented. A description of the SACO with changing parameter based on information entropy is then followed. The development of SACO model to deal with the construction project TCO problem will be proposed. In order to verify the efficiency and performance of the model, with the aid of a case study, simulation and performance comparison with other algorithms on the TCO problem are carried out. Finally, general remarks on this work and the direction of future research are pointed out.
2 Time-cost problem
The TCO problem is to select appropriate option for every activity to obtain the objectives of time and cost for a project. The time of a project T can be calculated according the following equation: Equation 1 where n – number of total activities of a project; t i k – duration of activity i when performing the kth option; x i k – index variable of activity i when performing the kth option: Equation 2 There are two parts of a project total cost: direct cost and indirect cost. Direct costs include expenses directly related to the production activities such as production costs, salaries for employees work in the manufactories, energy consumption for production, etc. Indirect costs include the all others expenses for current production period excluding the direct costs, which depends heavily upon the project duration, i.e. the longer the duration is, the higher the indirect cost is. Indirect costs include taxes, administration, personnel and security costs, etc.: Equation 3 where C – total cost of a project; D C i k – direct cost of activity i when performing the kth option; x i k – index variable of activity i when performing the kth option; t i – duration time of activity i; i c i k – indirect cost rate of a project.
The TCO of construction projects is a cornerstone in construction scheduling and planning efforts because it aims at finding the minimum cost to finish the project on time.
3 SACO with changing parameters
ACO is now a classical approach to solve combinatorial optimization problems, which were introduced by Dorigo (1992). One of its main ideas is the indirect communication among the individuals of a colony of agents, called ants. The principle of this method is based on the way ants search for food and find their way back to the nest. During the trips of ants, a chemical trail called pheromone is left on the ground. The role of pheromone is to guide the other ants towards the target point. For one ant, the path is chosen according to the quantity of pheromone. Recently, ACO has been proposed to solve combinatorial optimization problems, such as TSP (Dorigo and Gambardella, 1997), layout problem (Solimanpur et al., 2005) and flowshop scheduling problem (Rajendran and Ziegler, 2004).
3.1 Ant system for the TSP
We present the basic ACO algorithm, called ant system (AS), applied to the TSP. Here, we describe it in more detail as the following.
In the TSP is given a completely connected, undirected graph G(V, E ) with edge-weights. In this graph, the nodes V represent the cities, and the edge E weights represent the distances between the cities. The goal is to find a closed path in G that contains each node exactly once (henceforth called a tour) and whose length is minimal. Thus, the search space S consists of all tours in G. The objective function value f(s) of a tour s∈S is defined as the sum of the edge-weights of the edges that are in s. In other words, the notion of task of an ant changes from “choosing a path from the nest to the food source” to “constructing a feasible solution to the tackled optimization problem”.
Given an n-city TSP with distances d ij (i, j=1, 2, … , n), the artificial ants are distributed to these n cities randomly. Each ant will choose the next to visit according to the pheromone information remaining on the paths and heuristic information. However, there are two main differences between artificial ants and real ants:
the artificial ants have “memory”; they can remember the cities they have visited and therefore they would not select those cities again; and
the artificial ants are not completely “blind”; they know the distances between two cities and prefer to choose the nearby cities from their positions.
The pheromone trail is changed both locally and globally. Global updating is intended to reward edges belonging to shorter tours. Once artificial ants have completed their tours, the best ant deposits pheromone on visited edges; that is, on those edges that belong to its tour (the other edges remain unchanged). After one solution is completed meaning that the ant has traveled from city 1 to n, local pheromone updating rule as follows: Equation 5 where τ 0 – initial pheromone value on each edge and ρ – evaporation rate in the local updating process, ρ∈[0,1].
When an entire iteration is over, that is to say, all ants have completed their travels, the global pheromone should be updated. Obviously, the ants with shorter tours should leave more pheromone than those with longer tours. Therefore, the trail levels are updated as on a tour each ant leaves pheromone quantity given by Q/L k , where Q is a parameter and L k the length of its tour, respectively. On the other hand, the pheromone will evaporate as the time goes by. Then the updating rule of τ ij k could be written as follows: Equation 6 Equation 7 Equation 8 where t is the iteration number; ρ∈[0,1], is evaporation rate in the global updating; Δτ ij – the total increase value of pheromone on edge (i, j); Δτ ij k – the increase value of pheromone on edge (i, j) travelled by ant k.
3.2 Information entropy
Entropy is proposed to describe the state of the system. As a measure of uncertainty, information entropy was proposed by Shannon: Equation 9 where p i is the probability of the state, p i ≠0 and ∑ i=1 n P i =1. The information entropy has the following characteristics:
Symmetry characteristics: the value of the entropy does not depend on order of P 1, P 2, … , P n : Equation 10
Non-negative characteristics: Equation 11
Adding characteristics: the sum of the independent state's information entropy is equal to the total state's information entropy.
Extremism characteristics: when p i =1/n the information entropy reaches its biggest value ln n.
3.3 SACO with changing parameters based on information entropy
In the ACO, α and β are two parameters that control the relative weight of pheromone intensity and heuristic information. In the previous studies, they are constant, and should be selected by trailing, which reduces the efficiency of application of ACO. Furthermore, if the α is big and β is small in the beginning of iterations, premature convergence will occurrence, which influences the performance of the algorithm. Here, an algorithm with changing parameters will be proposed.
The information entropy is introduced to measure the uncertainty of the AS. The bigger the entropy is, the more uncertainty in the ants' choice of the next cities; the smaller entropy is, the more certainty in the ants' choice of the next cities. As a result, all of the ants find the best path and crawling the same way. At the same time, information entropy of the AS decreases to a small value, then the algorithm stops. The model is as follows.
Information entropy S=−k∑ i=1 n p i ln p i , and we take probability of ants selecting the next city, p ij k(t), into the equation. Information entropy of every ants S t can be gotten, so we definite mean entropy to measure the uncertainty of ants choosing path: Equation 12 where m – number of ants. Obviously, S¯ is a decreasing value: Equation 13 Equation 14 where b.d∈(0,1), are random numbers. So, α (t) and β (t) are controlled by the mean entropy, which ensure that in the beginning of the algorithm, α (t) is small, and with the iteration developing, α (t) is rising. At the same time, β (t) is big in the early stage, and with the iteration developing, β (t) is falling down. In other words, in the early stage, heuristic information τ ij (t) is weighted more important than pheromone information η ij (t), which makes the ants search solutions in a global feasible set and overcome the premature and trapping local best solutions. In the later stage, heuristic information τ ij (t) is weighted less important than pheromone information η ij (t) that is experience of self-learning, which guarantees stability of the algorithm and converging quickly.
For the stopping, we can define a small value of S¯. Because, in some cases, it is very difficult to decide the maximum iterative time for the complicated problems. We define when the information entropy is smaller than a given value (such as 0.01), the algorithm stops, and the solution is obtained.
4 SACO for TCO problem
4.1 TSP presentation for TCO
The basic idea of solving TCO problem with SACO is to transfer TCO into TSP. At the same time, a single objective would be gotten by combining the two objectives of time and cost. Finally, we calculate the optimal solution for TCO using SACO. TSP presentation for TCO can be shown as the Figure 1.
As the Figure 1 shows, A ij – the activity i performs the kth option; the activity 0 is a virtual activity that is the start of a project. A road starting at activity 0 and finishing at activity n is a project plan. Actually, the task of TCO problem is to find a road from activity 0 to activity n, and achieve an ideal balance between resource allocation costs and project schedule duration.
We set η ij (t) as a heuristic function, the visibility of city j from city i, which is always set as 1/d ij (d ij is the distance between activity i and j ). The heuristic information η ij (t) – the fitness function f(t), which is calculated by equation (18).
4.2 Modified adaptive weight approach
The TCO problem is a multiobjective optimization problem, which is characterized by a series of optimal solutions which may not be easily compared, as it is difficult and impossible to derive the “absolute best” solution that corresponds to all objectives as a basis for comparison. Each objective function may achieve its optimum at different points due to a lack of unified criteria with respect to the optima. Therefore, planners and managers shall apply their engineering judgment when selecting the “best” solution from a pool of optimal solutions along the Pareto front (Zheng et al., 2004, 2005). The Pareto front contains a set of no dominated solutions which are chosen as optimal because no objective can be further improved without sacrificing at least one other objective.
The method applies the MAWA proposed by Zheng et al. (2004) to solve the multiobjective problem, which is regarded as a very efficient approach.
Under the MAWA, the adaptive weights are formulated as the following four conditions.
For z t max ≠z t min and z c max ≠z c min : Equation 15 Equation 16 Equation 17 For z t max =z t min and z c max =z c min : Equation 18 For z t max =z t min and z c max ≠z c min : Equation 19 For z t max ≠z t min and z c max ≠z c min : Equation 20 where w t represents the weighting for total time; w c denotes the weighting for total cost; z t max and z c max are the maximal value for the objectives of total time and cost, respectively; z t min and z c min are the minimal value for the objectives of total time and cost, respectively; v t – value for the criterion of time; v c – value for the criterion of cost; v – value for the project; w t – adaptive weight for the criterion of time; and w c – adaptive weight for the criterion of cost.
4.3 SACO for TCO problem development
In the proposed model, an ant travels from activity 0 to n to find a solution with due consideration that an option must be selected by this ant according to the designated rule for each activity. For the selection probability, each ant k would generate a solution by traveling from the start to the end of a construction project implying that an ant can be regarded as a solution. The ant in activity i would select an option j using the pseudorandom proportional action choice rule as equation (4).
The heuristic information η ij (k) – the fitness function f(k), and the fitness function for the kth solution in the current iteration can be represented as follows: Equation 21 where γ – positive random number between 0 and 1, which is to avoid zero invalid value of the integrated; z t (k), z c (k) – value of the total direct and time of the kth solution, respectively.
The approach adopted in this study allows pheromone updating being performed according to both the local and global updating rules. Having completed a cycle, the pheromone value of the selected options τ ij as generated by the ant is updated according to the pheromone updating rules as equations (5)-(8). After all the ants finish their travels, the solutions are evaluated according to their fitness functions and the best ant (solution) is then selected.
In the SACO for TCO, the two parameters α, β are adjusting according to information entropy of the ants system S. So we do not need to select the two parameters at appropriate value by trialing.
For the stopping, we can define a small value of S¯, when S¯ is smaller than a given value (such as 0.01), the algorithm stops and the solution is obtained, which solves the problem that how to decide the maximum number of iterations for the complicated crit.
The flowchart of SACO for TCO is shown in Figure 2.
5 Case study and results comparison
In order to check the efficiency and performance of the algorithm, an example trialed by Zheng et al. (2005) using GA-based model and Thomas et al. (2008) using ACS-TCO model is fitted into prototype model. And the results have been compared by Thomas et al. (2008). In this study, to validate the efficiency of the SACO for TOC model proposed above, we carry out the simulation experiment with Matlab 7.0. The indirect cost is set at $1,500 per day. The parameters are set as follows: Q=200, ρ=0.2, b=0.2, d=0.1 as initial values and α, β are adjusting with mean information entropy; the number of ants m=20; when the mean information entropy S¯ is smaller than a given value 0.01, the algorithm stop and output the best solution.
The mean information entropy evolved over time as well as the evolution of the parameters alpha and beta as a line chart are shown in Figures 3-5. The efficiency of the SACO for TCO model can be confirmed by the iter-best solution for cost and time as shown in Figures 6 and 7, respectively. As Figures 6 and 7 show, the convergence is remarkable, and the value of total cost and time of the project are falling down with the iterations except for several points in which a slight upward deviation exhibits.
This indicates that SACO for TCO model is effective, which can generate the best solution in every round of iteration from the 120th iteration for cost and 140th for time iteration, respectively.
A comparison between the GA-based TCO model, ACS-TCO model, and SACO for TCO model is shown in Table I. We have done four rounds experiments, each round ran the three models for 100 times, and calculate the mean value of time and cost.
As Table I shows, it is apparent that the populations and iterations number of the SACO for TCO model are less than those of the GA-based model and ACS-TCO model. On the other aspect, the time and cost results for the case project, the SACO for TCO model can reach a more optimal cost value under less duration, and furthermore, the population in an iteration is less than the two models. It can be seen that all Pareto optimal solutions derived from the SACO algorithm are better than those of the other two approaches.
The Pareto fronts of the three approaches are shown in Figure 8. Through the comparison, it is draw the conclusion that the SACO for TCO model can achieve a better Pareto front than the GA-based model and ACS-TCO model.
6 Conclusion
In this paper the self-adaptive ACO has been introduced to model TCO problem so as to optimize the project objectives of duration and total cost. We use information entropy to measure the uncertainty of the AS. The bigger entropy is, the more uncertainty ants choose the path; the smaller entropy is, the more certainty ants choose the next cities. The parameters α and β have been controlled by the mean information entropy S¯, and when the S<0.01 the algorithm stops. In the early stage of the algorithm, α is small and β is big, which makes the ants search solutions in global feasible space and overcome the premature and trapping local best solutions. In the later stage, heuristic information τ ij (t) is weighted less important than pheromone information η ij (t) that is experience of self-learning, which guarantees stability of the algorithm and converging quickly. This improved algorithm has overcome the intrinsic deficiency of premature convergence. And we do not need to select the two parameters at appropriate value by trialing and appropriate iterations for stopping.
A computer simulation with Matlab 7.0 based on a prototype example has been carried out on the basis of SACO for TCO problem. The test results show that the SACO for TCO model can generate a more optimal cost under the same duration and achieve a better Pareto front than the GA-based model and ACS-TCO model. Therefore, the SACO can be regarded as a useful technique for solving construction project TCO problems. Despite that, further research on selection parameters Q and ρ should be conducted to further improve the robustness of the SACO for TCO model.
About the authors
Huimin Li is a Lecturer in the Department of Construction Engineering and Management at North China University of Water Resources and Electric Power, China. He holds a PhD in management science and engineering and he conducts research and publishes regularly in the area of construction engineering optimization. His interests also include a range of areas within the construction management discipline including project delivery system, procurement processes, risk management and schedule management. Huimin Li is the corresponding author and can be contacted at: [email protected]
Peng Li, a Lecturer, PhD, serves in College of Ideological and Political Education, North China University of Water Resources and Electric Power. He is the author and co-author of ten academic journal papers. His research interests include construction financing and engineering philosophy.
References
Ahmed, B.S. and Eldin, N.N. (2004), “Use of genetic algorithms in resource scheduling of construction project”, Journal of Construction Engineering and Management, Vol. 130 No. 6, pp. 869-877.
Burns, S. , Liu, L. and Feng, C. (1996), “The LP/IP hybrid method for construction time-cost trade-off analysis”, Construction Management and Economics, Vol. 14 No. 3, pp. 265-276.
Dorigo, M. (1992), “Optimization, learning and natural algorithms”, PhD thesis, Politecnico di Milano, Milan (in Italian).
Dorigo, M. and Gambardella, L.M. (1996), “Ant colonies for the traveling salesman problem”, TR/IRIDIA/1996-3, Université Libre de Bruxelles, Bruxelles.
Dorigo, M. and Gambardella, L.M. (1997), “Ant colony system: a cooperative learning approach to the traveling salesman problem”, IEEE Transactions on Evolutionary Computation, Vol. 1 No. 1, pp. 53-66.
Elbeltagi, E. , Hegazy, T. and Grierson, D. (2005), “Comparison among five evolutionary-based optimization algorithms”, Advanced Engineering Informatics, Vol. 19 No. 1, pp. 43-53.
Elmaghraby, S.E. (1993), “Resource allocation via dynamic programming in activity networks”, European Journal of Operational Research, Vol. 64 No. 2, pp. 199-215.
Eusuff, M.M. and Lansey, K.E. (2003), “Optimization of water distribution network design using the shuffled frog leaping algorithm”, Journal of Water Resources Planning and Management, Vol. 129 No. 3, pp. 210-225.
Feng, C. , Liu, L. and Burn, S. (1997), “Using genetic algorithms to solve construction time-cost trade-off problems”, Journal of Computing in Civil Engineering, Vol. 11 No. 3, pp. 184-189.
Feng, C. , Liu, L. and Burns, S. (2000), “Stochastic construction time-cost trade-off analysis”, Journal of Computing in Civil Engineering, Vol. 14 No. 2, pp. 117-126.
Fondahl, J.W. (1961), “A non-computer approach to the critical path method for the construction industry”, Technical Report No. 9, The Construction Institute, Department of Civil Engineering, Stanford University, Stanford, CA.
Kelly, J.E. Jr (1961), “Critical path planning and scheduling: mathematical basis”, Operation Research, Vol. 9 No. 3, pp. 167-179.
Kennedy, J. and Eberhart, R. (1995), “Particle swarm optimization”, Proceedings of IEEE International Conference on Neural Networks, Vol. IV, IEEE Press, Piscataway, NJ, pp. 1942-1948.
Li, Y. and Li, W. (2007), “Adaptive ant colony optimization algorithm based on information entropy: foundation and application”, Journal Fundamenta Informaticae, Vol. 77 No. 3, pp. 229-242.
Liu, L. , Burns, S. and Feng, C. (1995), “Construction time-cost trade-off analysis using LP/IP hybrid method”, Journal of Construction Engineering and Management, Vol. 121 No. 4, pp. 446-454.
Moscato, P. (1989), “On evolution, search, optimization, genetic algorithms and martial arts: towards memetic algorithms”, Technical Report No. 826, Caltech Concurrent Computation Program, California Institute of Technology, Pasadena, CA.
Moselhi, O. (1993), “Schedule compression using the direct stiffness method”, Canadian Journal of Civil Engineering, Vol. 20 No. 1, pp. 65-72.
Ng, S.T. and Zhang, Y. (2008), “Optimizing construction time and cost using ant colony optimization approach”, Journal of Construction Engineering and Management, Vol. 134 No. 9, pp. 721-728.
Ng, S.T. , Deng, M.Z. , Skitmore, R.M. and Lam, K.C. (2000), “A conceptual case-based decision module for mitigating construction delays”, International Journal of Construction Information Technology, Vol. 8 No. 2, pp. 1-20.
Rajendran, C. and Ziegler, H. (2004), “Ant-colony algorithms for permutation flowshop scheduling to minimize makespan/total flowtime of jobs”, European Journal of Operational Research, Vol. 155 No. 2, pp. 426-438.
Robinson, D.R. (1975), “A dynamic programming solution to cost-time tradeoff for CPM”, Management Science, Vol. 22 No. 2, pp. 158-166.
Siemens, N. (1971), “A simple CPM time-cost tradeoff algorithm”, Management Science, Vol. 17 No. 6, pp. 354-363.
Solimanpur, M. , Vrat, P. and Shankar, R. (2005), “An ant algorithm for the single row layout problem in flexible manufacturing systems”, Computers & Operations Research, Vol. 32 No. 3, pp. 583-598.
Wei, X. (2010), “Improved ant colony algorithm based on information entropy”, The 2010 International Conference on Computational and Information Sciences (ICCIS), Chendu, China, pp. 519-520.
Zhang, Y. and Ng, T. (2012), “An ant colony system based decision support system for construction time-cost optimization”, Journal of Civil Engineering and Management, Vol. 18 No. 4, pp. 580-589.
Zheng, D.X.M. , Ng, S.T. and Kumaraswamy, M.M. (2004), “Apply an ant colony optimization algorithm-based multiobjective approach for time-cost trade-off”, Journal of Construction Engineering and Management, Vol. 130 No. 2, pp. 168-176.
Zheng, D.X.M. , Ng, S.T. and Kumaraswamy, M.M. (2005), “Applying Pareto ranking and niche formation to genetic algorithm-based multiobjective time-cost optimization”, Journal of Construction Engineering and Management, Vol. 131 No. 1, pp. 81-91.
Acknowledgements
The authors thank the National Natural Science Foundation of China (Project No. 71302191), Department of Education of Henan province, Humanities and Social Science Projects (Project No. 2013-QN-028) and Basic Science Research Project of Henan province, China (Project No. 122300410029) for financially supporting this study.