The project scheduling problem with non-deterministic activities duration: A literature review

Ortiz-Pimiento, Nestor Raul; Diaz-Serna, Francisco Javier Article The project scheduling problem with non-deterministic activities duration: A literature review Journal of Industrial Engineering and Management (JIEM) Provided in Cooperation with: The School of Industrial, Aerospace and Audiovisual Engineering of Terrassa (ESEIAAT), Universitat Politècnica de Catalunya (UPC) Suggested Citation: Ortiz-Pimiento, Nestor Raul; Diaz-Serna, Francisco Javier (2018) : The project scheduling problem with non-deterministic activities duration: A literature review, Journal of Industrial Engineering and Management (JIEM), ISSN 2013-0953, OmniaScience, Barcelona, Vol. 11, Iss. 1, pp. 116-134, https://doi.org/10.3926/jiem.2492 This Version is available at: https://hdl.handle.net/10419/188852 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Journal of Industrial Engineering and Management JIEM, 2018 - 11(1): 116-134 - Online ISSN: 2013-0953 - Print ISSN: 2013-8423 https://doi.org/10.3926/jiem.2492 The Project Scheduling Problem with Non-Deterministic Activities Duration: A Literature Review Nestor Raul Ortiz-Pimiento1, Francisco Javier Diaz-Serna2 1Universidad Industrial de Santander (Colombia) 2Universidad Nacional de Colombia (Colombia) [email protected], [email protected] Received: November 2017 Accepted: February 2018 Abstract: Purpose: The goal of this article is to provide an extensive literature review of the models and solution procedures proposed by many researchers interested on the Project Scheduling Problem with nondeterministic activities duration. Design/methodology/approach: This paper presents an exhaustive literature review, identifying the existing models where the activities duration were taken as uncertain or random parameters. In order to get published articles since 1996, was employed the Scopus database. The articles were selected on the basis of reviews of abstracts, methodologies, and conclusions. The results were classified according to following characteristics: year of publication, mathematical representation of the activities duration, solution techniques applied, and type of problem solved. Findings: Genetic Algorithms (GA) was pointed out as the main solution technique employed by researchers, and the Resource-Constrained Project Scheduling Problem (RCPSP) as the most studied type of problem. On the other hand, the application of new solution techniques, and the possibility of incorporating traditional methods into new PSP variants was presented as research trends. Originality/value: This literature review contents not only a descriptive analysis of the published articles but also a statistical information section in order to examine the state of the research activity carried out in relation to the Project Scheduling Problem with non-deterministic activities duration. Keywords: project scheduling, random duration, uncertain duration, stochastic duration, task duration 1. Introduction The project scheduling problem (PSP) is a generic name for every problem that focus on the optimizing of the project duration, the allocation of the project resources, the estimated project costs, and the project’s cash flow, among others. In order to achieve these main goals, PSP aim to generate a sequence of activities, organized according to a decision criterion to give a proper solution to problem addressed. Considered as an NP-hard problem (Lancaster & Ozbayrak, 2007), PSP models usually have a deterministic approach. However, some models might include uncertainty or randomness in the input parameters, corresponding to a non-deterministic direction. This article presents an exhaustive literature review, identifying the existing models where the activities duration were taken as uncertain or random parameters. In order to get published articles since 1996, was employed the Scopus database. -116- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 The PSP with non-deterministic activities duration includes at least the following versions: Basic Project Scheduling Problem (PSP), Resource-Constrained Project Scheduling Problem (RCPSP), Resource-Constrained Project Scheduling Problem with multiple objectives (Multi-Objective RCPSP), Multi-Mode Resource-Constrained Project Scheduling Problem (MRCPSP), Resource-Constrained Multiple Project Scheduling Problem (RCMPSP), and Time/Cost Trade-off Problem (TCTP). Solution procedures presented in literature, were classified by Brčić, Kalpic and Fertalj (2012) according to three specific approaches: a predictive strategy, a proactive strategy and a reactive strategy: The predictive approach take the average activities duration as input data and creates a project baseline. The problem is solved as a deterministic problem. The proactive approach takes into account the variation of the activities duration, and generate a robust baseline for the project. The robustness concept indicates that the linebase will require little changes when the project risks appear. Three types of proactive solution were identified in literature (Brčić et al., 2012): redundancy based methods, robust scheduling methods, and contingent scheduling methods. •Redundancy based methods provide additional time for the activities in order to face the risks. The additional time can be incorporated for extending the original duration of each task or inserting buffers. •Robust scheduling methods proposes a baseline for the project through optimization model whose objective function is based on a robustness measure. The most common robustness measure can be obtained by the weighted sum of the absolute deviation between the planned and realized activity start times (Van De Vonder, Demeulemeester, Herroelen & Leus 2006). •Contingent scheduling methods generates more than one baseline for the project. The risks are analyzed previously and a baseline is created for each possible disruption, then there will be alternative action plans The reactive approach creates a strategy to re-schedule the original schedule when an unexpected event takes place. The re-schedule can be carried out by re-schedule the whole original sequence (using an optimization process) or re-schedule a little part of the network with the following strategies: •The Right Shift Rule, where the delayed tasks should be to move toward the right consuming their slack times. If the slack time is not enough, the actions to reduce the subsequent activities duration are necessary. •Activity crashing, where the subsequent activities duration should be reduced. This action implies an increase in the amount of resources and it generates additional costs. The Time/Cost Trade-off Problem (TCTP) allow to identify the activities that must be intervened. •Activities overlapping, where the types of precedence should be redefined. If a project is behind schedule and the policy shows that the activities only start when its predecessors end (end-start precedence), the new policy can suggest a change in the type of precedence to the stat-start or end-end precedence. The relationship between proactive and reactive scheduling is very strong, since a project baseline requires a reactive strategy to face the disruptions that appear during the project execution. However, Rostami, Creemers and Leus (2017) present an alternative reactive strategy, called purely reactive strategy, on-line strategy, and also known as stochastic scheduling, one that doesn’t generate a project baseline, requiring the design of a policy or decision rule to schedule the project activities. On Section 2, the PSP with non-deterministic activities duration is addressed, and relevant statistical information created from the literature review process is also presented. Sections 3 and 4, contains a description of the contributions and solutions techniques proposed by outstanding researchers. Section 5, some research trends are identified as a result of the statistical analysis performed, supplementary to the one reviewed on Section 2. Finally, conclusions are given in Section 6. -117- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 2. Project Scheduling Problem with Non-Deterministic Activities Duration This section presents the PSP with non-deterministic activities duration. The documents reviewed were classified according to following characteristics: year of publication, mathematical representation of the activities duration, solution techniques applied, and type of problem solved. 2.1. According to the Year of Publication Figure 1 shows the number of articles published between 1996 and 2017. Since 2007, the great interest of researchers on the PSP with non-deterministic activities duration is remarkable. Figure 1. Number of published articles per year During 1996-2017, the total number of scientific articles published was 255. However, in 2018, 3 additional articles have been identified. 2.2. According to the Mathematical Representation of the Activities Duration On Table 1, it is observed that the most common way to represent the activities duration is through random variable. However, depiction through uncertainty measurements, such as fuzzy numbers, was employed in the 32.17% of the reviewed articles. Mathematical representation Number of publications Percentage Random variable 169 65.50% Uncertainty measurement 83 32.17% Combination: Randomness and uncertainty 6 2.33% Total 258 100.00% Table 1. Mathematical representation of the activities duration Additionally, a small percentage of cases, where the authors employed fuzzy numbers and random variables, is identified. 2.3. According to the Solution Techniques Applied Table 2 indicates that traditional meta-heuristics and procedures specifically designed by researchers, were the solution techniques most applied to solve PSP with non-deterministic activities duration. Just like -118- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 meta-heuristics, the Special Procedures (SP) can generate high-quality solutions with small computational effort. SP are supported in priority rules, stage-by-stage analysis, or simple search algorithms. The exact methods, are also applied on the 21.32% of the reviewed articles, especially the Branch and Bound technique (B&B) and Dynamic Programming. Solution technique Number of publications Percentage Traditional Meta-heuristics 71 27.52% Special Procedures (SP) 60 23.26% Exact Methods 55 21.32% Others algorithms 32 12.40% Critical Chain and Buffer Management (CC/BM) 27 10.47% Networks 13 5.04% Total 258 100.00% Table 2. Solution techniques applied by researchers 2.4. According to the Type of Problem The results presented on Table 3 indicate that RCPSP was the problem most frequently analyzed by researchers. This type of problem appears on the 46.12% of the published articles. However, the basic Project Scheduling Problem (PSP) was also highlighted, analyzed in the 27.91% of the reviewed articles. Others variants of the problem were reported with little frequency. Type of Problem Number of publications Percentage RCPSP 119 46.12% PSP 72 27.91% TCTP 16 6.20% Multi-objective RCPSP 16 6.20% MRCPSP 11 4.26% RCMPSP 9 3.49% Others 15 5.81% Total 258 100.00% Table 3. Type of problem analyzed by researchers 3. Researchers and Relevant Articles Articles compiled in this literature review have been presented by more than 300 researchers of different countries. Roel Leus with 12 articles and Hua Ke with 10, were the authors with the highest number of publications. On the other hand, it is also important to mention the two most cited articles: “A Simulation-Based Process Model for Managing Complex Design Projects” (Cho & Eppinger, 2005), which has 223 citations, and presents a heuristic to schedule sequential, parallel or overlap activities; and “An investigation of buffering techniques in critical chain scheduling” (Tukel, Rom & Eksioglu, 2006), which has 111 citations, and presents two methods to determine the Feed Buffer Size under Critical Chain concept. Other highlights articles can be seen in Table 4. -119- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 Article Number of citations Applied techniques Type of problem analyzed (Cho & Eppinger, 2005) 223 Special Procedure RCPSP (Tukel, Rom & Eksioglu, 2006) 111 CC/BM RCPSP (Van De Vonder, Demeulemeester, Herroelen & Leus, 2005) 108 CC/BM PSP (Long & Ohsato, 2008) 102 CC/BM RCPSP (Van De Vonder, Demeulemeester & Herroelen, 2008) 96 Specials Procedures RCPSP (Wang, 2004) 92 Genetic Algorithm RCPSP (Golenko-Ginzburg & Gonik, 1997) 86 Special Procedures RCPSP (Gutjahr, Strauss & Wagner, 2000) 69 Branch and Bound PSP (Subramanian, Pekny & Reklaitis, 2001) 67 Branch and Bound RCMPSP (Chtourou & Haouari, 2008) 64 Special Procedure RCPSP Table 4. Ten most cited articles published 3.1. Roel Leus’s Contributions Roel Leus proposed a mathematical model for the allocation of resources, which protects the project baseline against the variation on the length of the activities. The B&B algorithm was applied as solution technique (Leus & Herroelen, 2002). In 2005, Leus was co-author of a remarkable article (Van De Vonder et al., 2005), which contrasted the solutions quality of three different procedures to schedule projects: the original version of Critical Chain and Buffer Management (CC/BM), a modified version of CC/BM, and the Adapted Floating Factor method (ADFF). Later, Leus proposed an algorithm to minimize the Stability Cost function. Current, this objective function is considered an important measure of robustness (Herroelen & Leus, 2004). In 2007 he designed an GRASP algorithm to solve the RCPSP which had greater performance than the previous algorithms (Ballestin & Leus, 2007). In 2008 he applied a backward stochastic dynamic programming recursion to maximize the Net Present Value (Creemers, Leus, De Reyck & Lambrecht, 2008; Creemers, Leus & Lambrecht, 2010). Additionally, Leus developed a new model with decision nodes to evaluate the viability of research projects (Creemers, Leus & De Reyck, 2010). In 2013, he analyzed the high uncertainty in the duration of activities and proposed two algorithms based on scenarios relaxation (Artigues, Leus & Talla Nobibon, 2013). Recently, Leus contributed to the development of a new class of policies for project scheduling, where sequencing decisions are made in a pre-processing phase, while other decisions are made during the execution of the project (Ashtiani, Leus & Aryanezhad, 2011; Leus, Rostami & Creemers, 2015; Rostami et al., 2017). 3.2. Hua Ke’s Contributions In most of his articles, Hua Ke has utilized hybrid algorithms that integrate techniques as simulation process, schedule generation schemes (SGS), and Genetic Algorithms (GA). In 2005, a hybrid algorithm was utilized to solve a PSP with an objective function that evaluated the total cost under some completion time limits (Ke & Liu, 2005). Later, he incorporated the Fuzzy Random Variable concept to model the uncertainty on activities duration (Ke & Liu, 2007). After, in 2009 and 2010, Hua Ke tackled the Time/Cost Trade-off Problem (TCTP) (Ma & Ke, 2009; Ke, Ma, Gao & Xu, 2010), where the activities durations were assumed as fuzzy variables. In 2012, he analyzed again the previous algorithm, but in this case, the activities durations were assumed as fuzzy random variables (Ke, Ma & Ma, 2012). Additionally, Ke proposed a model called Stochastic Time-dependent Time/Cost Trade-off Problem, where the activities are executed according to original scheduling, but random delays can appear (Ke et al., 2012). In 2015, he proposed a model to tackle the PSP, where the activities durations were assumed as fuzzy and random variables (Ke, Liu & Tian, 2015). Recently, Hua Ke explored the RCPSP and developed efficient algorithms to solved it (Wang, Huang & Ke, 2015; Ma, Che, Huang & Ke, 2016). Finally, in 2017, he analyzed the Resource -120- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 Leveling Problem with uncertain durations and a deadline constraint, and proposed an algorithm that integrated a special SGS and a Distribution Estimation Algorithm (Ke & Zhao, 2017). 4. Solution Techniques PSP with non-deterministic activities duration has been solved through the development and application of some solution techniques, such as the traditional meta-heuristics methods, special procedures (SP), exact methods, and Critical Chain and Management Buffers (CC/BM). In the following paragraphs, the main articles are mentioned and classified according to the solution technique applied. 4.1. Traditional Meta-Heuristics Genetic algorithm (GA) was the Meta-heuristic method most frequently applied in scientific literature for the different versions of PSP with non-deterministic activities duration. However, various other Meta-heuristics as particle swarm optimization (PSO), Tabu search (TS), Bee Colony, Ant Colony, Greedy Algorithms, Simulated annealing (SA), and Distribution Estimation Algorithm (DEA), have been used as solution techniques (see Table 5). Meta-heuristic applied Type of problem Total PSP RCPSP TCTP Multi-objective RCPSP MRCPSP RCMPSP Others GA 6 14 5 10 2 1 38 PSO 3 4 1 8 TS 1 1 1 1 4 Bee colony 1 1 2 Ant colony 1 1 Greedy algorithms 2 1 3 SA 1 1 2 4 DEA 2 1 3 Integrated algorithms 1 4 1 1 1 8 Table 5. Traditional meta-heuristic applied 4.1.1. Genetic Algorithms Hybrid algorithms were developed by Hua Ke to solve the PSP. These algorithms integrated GA with others solution techniques, as simulation process and schedule generation schemes (SGS) (Ke & Liu, 2005; Ke & Liu, 2007; Ke et al., 2012; Ke et al., 2015). Recently, Wang and Ning designed a new GA to solve an uncertain chance-constrained programming model (Wang & Ning, 2017); and on the other hand, Ji and Yao applied GA to solve an uncertain multi-objective programming model where the duration times and the resource allocation times of the activities were described as uncertain variables (Ji & Yao, 2017). In RCPSP, the use of GA, widely exceed the number of cases reported to other PSP versions. Wang (2004) applied GA to the new product development project case; Liu, Yung, and Ip (Liu, Yung & Ip, 2007) and Liu (Liu, Zhao, Zhang & Du, 2007), found solutions to reduce the project duration. Additionally, Huang developed a procedure that integrate GA with fuzzy simulation (Huang, Ding, Wen & Cao, 2009; Wang & Huang, 2010) and another that integrated GA with a fuzzy parallel schedule generation scheme (Huang, Shou & Zhang, 2011). In 2010, Zhao, You and Zuo (2010) designed a procedure to solve RCPSP by incorporating buffers into the project network. In 2011, Masmoudi and Haït (2011) developed an GA for Resource Leveling Problem (LRP). Later, Mogaadi and Chaar (2015) designed a solution procedure that combines GA and the Forward Backward Improvement heuristic (FBI). Recently, Chen, Xiong, and Zhou (2016) applied the Resilience concept to measure the schedule's ability to absorb possible perturbation in the project. -121- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 A Bi-objective RCPSP was solved by Zhang (2015) applying a Non-dominated Sorting Genetic Algorithm (NSGA II). This model, minimizes the project duration and maximizes the robustness of the solution. NSGA II was also used by Tabrizi and Ghaderi (2016), and Ghoddousi, Ansari and Makui (2017). New algorithms allow to combine GA and other heuristics in order to improve their performance: Pan, Willis and Yeh (2001) combined GA and Tabu Search (TS); Wang et al. (2015) used a hybrid algorithm integrating GA and Schedule Generator Scheme (SGS); Ma et al. (2016) integrated GA with the 99-method. 4.1.2. Particle Swarm Optimization On the other hand, PSO was another important meta-heuristic used to solve the PSP with non-deterministic activities duration. Chen, Xiao and Lu (2011) and Gan and Xu (2015) analyzed the MRCPSP and applied PSO in order to minimize the length of the project. Xu and Feng (2014) proposed an integrated model with multiples modes and multiples objectives, and Yaghoubi, Noori and Azaron (2015), solved a multi-objective continuous-time problem. In 2014, Ma and Xu (2014) applied PSO to solve a multi-criteria multi-project RCPSP. In this article, the project owner seeks to maximize profits whereas the contractor attempts to minimize cost. Finally, PSO was utilized by Zhang to solve the RCPSP with multiple objectives or multiple modes, where the contractor is the Upper Level Decision Maker and the outsourcing partner is the Lower Level Decision Maker (Zhang, 2014; Zhang, Liu, Zhou & Chen, 2015; Zhang & Xu, 2015). 4.1.3. Integrated Algorithms Integration of others algorithms has also been reported in the literature: Kerkhove and Vanhoucke (2017) combined Simulated Annealing (SA) and a Dedicated Algorithm; Masmoudi and Haït (2013) combined GA and Greedy Algorithm; and Kumar and Srivastava (2014), combined SA and Multi-objective GA. 4.2. Special Procedures (SP) Special Procedures can generate high-quality solutions with small computational effort. These procedures can be supported in priority rules, simulation process, stage-by-stage analysis or simple search algorithms. Table 6 shows articles sorted according type of problem analyzed. Special procedure based on Type of problem Total PSP RCPSP TCTP Multi-objective RCPSP MRCPSP RCMPSP Others Priority rules 1 8 1 10 Simulation process 7 1 1 9 Stage-by-stage analysis 1 6 1 8 Simple search algorithms 1 4 5 Others 9 16 2 1 28 Table 6. Special procedures applied 4.2.1. Special Procedures Based on Priority Rules Golenko-Ginzburg and Gonik designed algorithms to select the activities that will be scheduled in the decision points located into the project network. These algorithms ere designed to solve the RCPSP and to minimize the expected project duration, take into account the available resources (Golenko-Ginzburg & Gonik, 1997; GolenkoGinzburg & Gonik, 1998). In 2007, Rabbani, Fatemi Ghomi, Jolai and Lahiji (2007) identified points decision, utilized the backward pass method to obtain the start time of each activity, and presented a new heuristic to determine the finish time. Rabbani, Baradaran, Fatemi-Ghomi, and Hashemin (2008) tackled the RCPSP and developed a new constructive heuristic rule based on Time Criticality Index (TCI) and Resource Criticality Index (RCI). Fu, Lau and Xiao (2008), -122- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 presented the RCPSP with minimum and maximum time lags (RCPSP/max) model, and applied a Parcial Order Schedule (POS) to solve the problem. Recently, Knyazeva, Bozhenyuk and Rozenberg (2015) proposed a heuristic based on priority rule to meet the project deadline. 4.2.2. Special Procedures Based on Simulation Process Pet-Edwards built a RCPSP model and developed a special procedure supported on simulation processes (Pet-Edwards & Mollaghesemi, 1996; Fernandez, Armacost & Pet-Edwards, 1998). Golenko-Ginzburg, Gonik and Laslo (2003), proposed a simulation process to solve a network project that include both alternative deterministic decision nodes and alternative branching nodes with probabilistic outcome. Later, Golenko-Ginzburg, Gonik and Baron (2006), analyzed simultaneous projects of PERT type and some resource scheduling models. Blaszczyk and Nowak (2009) solved the TCTP and created a new procedure based on computer simulation and interactive approach. The procedure uses simulation experiments to evaluate decision alternatives, and an interactive technique to obtain the final solution. The procedure uses stochastic dominance rules for comparing decision alternatives. Recently, a simulation process was developed for activities crashing and to reduce the duration of the project (Subhy, Georgy & Ibrahim, 2014). 4.2.3. Special Procedures Based on Stage-by-Stage Analysis Mizuyama (2006) formulated the PSP as a multi-stage probabilistic decision-making process. The model allow to maximize the project's output quality, and to meet the deadline. Chtourou and Haouari (2008) developed a two-stages procedure: in the first stage, the procedure uses priority rules to minimize the makespan; in the second stage, the procedure selects the best solution based on a robustness indicator. In 2010, Hazir, Haouari and Erel (2010), presented the TCTP and designed a two-phases procedure: in the first phase the minimum required budget is determined, and in the second phase, the buffer size is maximized. Lambrechts, Demeulemeester and Herroelen (2011), designed a procedure to insert extra time due to disruptions caused by the unavailabilities of resources. The procedure includes two phases: the resources assignation and the insertion of extra time. Leus et al. (2015), developed a procedure to find schedule policies. Initially, priority list are generated by a GRASP algorithm, and subsequently, the precedence constraints are created by a GA. Recently, in 2016, Tseng and Ko (2016) presented a two-stages procedure: in the first stage, a scenario tree is generated, and in the second stage, the worst branches are eliminated using the Expected Utility-Entropy criteria. 4.3. Exact Methods B&B is the most representative exact technique used to solve PSP with non-deterministic activities duration (see Table 7). However, this technique has frequently been applied to project networks containing less than 60 activities, due to the high computational effort required to solve large size problems. Exact method applied Type of problem Total PSP RCPSP TCTP Multi-objective RCPSP MRCPSP RCMPSP Others Branch and Bound 8 9 2 1 1 21 Dynamic Programming 5 6 1 2 14 Stochastic programming 2 2 2 6 Others 6 3 2 2 1 14 Table 7. Exact methods applied -123- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 Ke, H., Ma, W., Gao, X., & Xu, W. (2010). New fuzzy models for time-cost trade-off problem. Fuzzy Optimization and Decision Making, 9(2), 219-231. https://doi.org/10.1007/s10700-010-9076-z Ke, H., Ma, W., & Ma, J. (2012). Solving project scheduling problem with the philosophy of fuzzy random programming. Fuzzy Optimization and Decision Making, 11(3), 269-284. https://doi.org/10.1007/s10700-012-9133-x Kerkhove, L., & Vanhoucke, M. (2017). Optimised scheduling for weather sensitive offshore construction projects. Omega, 66, Part A, 58-78. https://doi.org/10.1016/j.omega.2016.01.011 Klerides, E., & Hadjiconstantinou, E. (2010). A decomposition-based stochastic programming approach for the project scheduling problem under time/cost trade-off settings and uncertain durations. Computers and Operations Research, 37(12), 2131-2140. https://doi.org/10.1016/j.cor.2010.03.002 Klerides, E., & Hadjiconstantinou, E. (2015). The Stochastic Discrete Time-Cost Tradeoff Problem with Decision-Dependent Uncertainty. In Handbook on Project Management and Scheduling, 2, 781-809. https://doi.org/10.1007/978-3-319-05915-0 Knyazeva, M., Bozhenyuk, A., & Rozenberg, I. (2015). Resource-constrained Project Scheduling Approach Under Fuzzy Conditions. Procedia Computer Science, 77, 56-64. https://doi.org/10.1016/j.procs.2015.12.359 Kokkaew, N., & Chiara, N. (2010). Modelling completion risk using stochastic critical path‐envelope method: a BOT highway project application. Construction Management and Economics, 28(12), 1239-1254. https://doi.org/10.1080/01446193.2010.521755 Kumar, B., & Srivastava, S. (2014). Integrated Fuzzy - HMH for project uncertainties in time - cost tradeoff problem. Applied Soft Computing Journal, 21, 320-329. https://doi.org/10.1016/j.asoc.2014.03.035 Lamas, P., & Demeulemeester, E. (2014). A purely proactive scheduling procedure for the resource-constrained project scheduling problem with stochastic activity durations. Belgium: Faculty of Business and Economics KU Leuven. https://doi.org/10.1007/s10951-015-0423-3 Lambrechts, O., Demeulemeester, E., & Herroelen, W. (2011). Time slack-based techniques for robust project scheduling subject to resource uncertainty. Annals of Operations Research, 186(1), 443-464. https://doi.org/10.1007/s10479-010-0777-z Lancaster, J., & Ozbayrak, M. (2007). Evolutionary algorithms applied to project scheduling problems–a survey of the state-of-the-art. International Journal of Production Research, 45(2), 425-450. https://doi.org/10.1080/00207540600800326 Lee, D.-E. (2005). Probability of Project Completion Using Stochastic Project Scheduling Simulation. Journal of Construction Engineering and Management, 131(3), 310-318. https://doi.org/10.1061/(ASCE)0733-9364(2005)131:3(310) Leus, R., & Herroelen, W. (2002). Stability and resource allocation in project planning. IIE Transactions, 36(7). https://doi.org/10.1080/07408170490447348 Leus, R., Rostami, S., & Creemers, S. (2015). New Benchmark Results for the Problem Constrained Project Scheduling. International Conference on Industrial Engineering and Engineering Management, 43(11), 1485-1492. https://doi.org/10.1287/mnsc.43.11.1485 Li, H., & Womer, N.K. (2015). Solving stochastic resource-constrained project scheduling problems by closed-loop approximate dynamic programming. European Journal of Operational Research, 246(1), 20-33. https://doi.org/10.1016/j.ejor.2015.04.015 Li, X., Wang, W., & Zeng, X. (2010). The study on resource constraint project scheduling problem under stochastic circumstances. In 2nd Conference on Environmental Science and Information Application Technology, ESIAT 2010, 3, 648-651. https://doi.org/10.1109/ESIAT.2010.5568738 Liberatore, M.J. (2008). Critical Path Analysis With Fuzzy Activity Times. IEEE Transactions on Engineering Management, 55(2), 329-337. https://doi.org/10.1109/TEM.2008.919678 -130- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 Liu, D., Chen, J., & Peng, W. (2012). A new buffer setting method based on activity attributes in construction engineering Deyin Liu 1. Applied Mechanics and Materials, 177, 3274-3281. https://doi.org/10.4028/www.scientific.net/AMM.174-177.3274 Liu, S., Yung, K.L., & Ip, W.H. (2007). Genetic local search for resource-constrained project scheduling under uncertainty. International Journal of Information and Management Sciences, 18(4), 347-363. Liu, Y., Zhao, S.L., Zhang, X.P., & Du, G.Q. (2007). A GA-based approach for solving fuzzy project scheduling. In Proceedings of the Sixth International Conference on Machine Learning and Cybernetics, ICMLC, 6, 3153-3156. https://doi.org/10.1109/ICMLC.2007.4370690 Long, L.D., & Ohsato, A. (2008). Fuzzy critical chain method for project scheduling under resource constraints and uncertainty. International Journal of Project Management, 26, 688-698. https://doi.org/10.1016/j.ijproman.2007.09.012 Ma, W., Che, Y., Huang, H., & Ke, H. (2016). Resource-constrained project scheduling problem with uncertain durations and renewable resources. International Journal of Machine Learning and Cybernetics, 7(4), 613-621. https://doi.org/10.1007/s13042-015-0444-4 Ma, W., & Ke, H. (2009). Modeling time-cost trade-off problem with fuzzy activity duration times. In 4th International Conference on Cooperation and Promotion of Information Resources in Science and Technology, COINFO, 344-347. https://doi.org/10.1109/COINFO.2009.44 Ma, Y., & Xu, J. (2014). A novel multiple decision-maker model for resource-constrained project scheduling problems. Canadian Journal of Civil Engineering, 511(April), 500-511. https://doi.org/10.1139/cjce-2013-0232 Mansoorzadeh, S., & Mohd Yusof, S. (2011). Reliable project scheduling with combination of risk management and critical chain schedule. In IEEE Student Conference on Research and Development, 442-447. https://doi.org/10.1109/SCOReD.2011.6148780 Masmoudi, M., & Haït, A. (2011). A GA-based fuzzy resource leveling optimization for helicopter maintenance activity. Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2011 and French Days on Fuzzy Logic and Applications, LFA, 1(1), 665-672. Masmoudi, M., & Haït, A. (2013). Project scheduling under uncertainty using fuzzy modelling and solving techniques. Engineering Applications of Artificial Intelligence, 26(1), 135-149. https://doi.org/10.1016/j.engappai.2012.07.012 Mizuyama, H. (2006). A time quality tradeoff problem of a project with nonstandardized activities. In 36th International Conference on Computers and Industrial Engineering, ICC and IE, 3039-3049. https://doi.org/10.1057/jors.1988.132 Mogaadi, H., & Chaar, B.F. (2015). Scenario-based evolutionary approach for robust RCPSP. In Proceedings of the second international Afro-European conference for industrial advancement AECIA 2015, 45-55. https://doi.org/10.1007/978-3-319-29504-6 Nelson, R.G., Azaron, A., & Aref, S. (2016). The use of a GERT based method to model concurrent product development processes. European Journal of Operational Research, 250(2), 566-578. https://doi.org/10.1016/j.ejor.2015.09.040 Pan, H., Willis, R.J., & Yeh, C. (2001). Resource-constrained Project Scheduling with Fuzziness. In Advances in Fuzzy Systems and Evolutionary Computation, 173-179. World Scientific and Engineering Society Press. Pet-Edwards, J.J., & Mollaghesemi, M. (1996). A simulation and genetic algorithm approach to stochastic research\nconstrained project scheduling. Southcon/96 Conference Record, 333-338. https://doi.org/10.1109/SOUTHC.1996.535089 Rabbani, M., Baradaran, S., Fatemi Ghomi, S.M.T., & Hashemin, S.S. (2008). Development of a Constructive Heuristics Rule for Constrained Resource Allocation in Stochastic Networks. Journal of Applied Sciences, 8(21), 3917-3923. -131- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 Rabbani, M., Fatemi Ghomi, S.M.T., Jolai, F., & Lahiji, N.S. (2007). A new heuristic for resource-constrained project scheduling in stochastic networks using critical chain concept. European Journal of Operational Research, 176(2), 794808. https://doi.org/10.1016/j.ejor.2005.09.018 Ramat, E., Lente, C., Slimane, M., Tacquard, C., & Venturini, G. (1996). Stochastic Project Scheduling Based on Time Lag. In IEEE International Conference on Systems, Man and Cybernetics, 2916-2921. Roghanian, E., Alipour, M., & Rezaei, M. (2017). An improved fuzzy critical chain approach in order to face uncertainty in project scheduling. International Journal of Construction Management, 1(February), 1-13. https://doi.org/10.1080/15623599.2016.1225327 Rostami, S., Creemers, S., & Leus, R. (2017). New strategies for stochastic resource-constrained project scheduling. Journal of Scheduling, 20(1), 1-17. https://doi.org/10.1007/s10951-016-0505-x Said, S.S., & Haouari, M. (2015). A hybrid simulation-optimization approach for the robust Discrete Time/Cost Trade-off Problem. Applied Mathematics and Computation, 259, 628-636. https://doi.org/10.1016/j.amc.2015.02.092 Saihjpal, V., & Singh, S.B. (2014). New Placement Strategy for Buffers in Critical Chain. In Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30, 2012, 429-436. https://doi.org/10.1007/978-81-322-1602-5 Schmidt, C.W., & Grossmann, I.E. (1996). A mixed integer programming model for stochastic scheduling in new product development. Computers & Chemical Engineering, 20(96), S1239-SI244. Shi, Q., & Gong, T. (2009). An improved project buffer sizing approach to critical chain management under resources constraints and fuzzy uncertainty. In International Conference on Artificial Intelligence and Computational Intelligence, 486-490. https://doi.org/10.1109/AICI.2009.192 Sobel, M.J., Szmerekovsky, J.G., & Tilson, V. (2009). Scheduling projects with stochastic activity duration to maximize expected net present value. European Journal of Operational Research, 198(3), 697-705. https://doi.org/10.1016/j.ejor.2008.10.004 Subhy, E.S., Georgy, M.E., & Ibrahim, M.E. (2014). Incorporating Uncertainty into Project Schedule Crashing: An Algorithm. In Proceedings of the 31st International Symposium on Automation and Robotics in Construction, ISARC, 404_409. Subramanian, D., Pekny, J.F., & Reklaitis, G.V. (2001). A simulation-optimization framework for research and development pipeline management. AIChE Journal, 47(10), 2226-2242. https://doi.org/10.1002/aic.690471010 Tabrizi, B.H., & Ghaderi, S.F. (2016). A robust bi-objective model for concurrent planning of project scheduling and material procurement. Computers & Industrial Engineering, 98, 11-29. https://doi.org/10.1016/j.cie.2016.05.017 Tereso, A.P., Araujo, M., & Elmaghraby, S.E. (2004). Adaptive resource allocation in multimodal activity networks. International Journal of Production Economics, 92(1), 1-10. https://doi.org/10.1016/j.ijpe.2003.09.005 Tian, W., Xu, J., & Fu, Z. (2017). On the choice of baseline schedules for the discrete time / resource trade-off problem under stochastic environment. Journal of Difference Equations and Applications, 6198(September), 1-11. https://doi.org/10.1080/10236198.2016.1155566 Tseng, C., & Ko, P. (2016). Measuring schedule uncertainty for a stochastic resource-constrained project using scenario-based approach with utility-entropy decision model. Journal of Industrial and Production Engineering, 1015(May), 1-10. https://doi.org/10.1080/21681015.2016.1172522 Trietsch, D., & Baker, K.R. (2012). PERT 21: Fitting PERT/CPM for use in the 21st century. International Journal of Project Management, 30(4), 490-502. https://doi.org/10.1016/j.ijproman.2011.09.004 Tukel, O.I., Rom, W.O., & Eksioglu, S.D. (2006). An investigation of buffer sizing techniques in critical chain scheduling. European Journal of Operational Research, 172, 401-416. https://doi.org/10.1016/j.ejor.2004.10.019 -132- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 Van De Vonder, S., Demeulemeester, E., Herroelen, W., & Leus, R. (2005). The use of buffers in project management: The trade-off between stability and makespan. Int. J. Production Economics, 97, 227-240. https://doi.org/10.1016/j.ijpe.2004.08.004 Van De Vonder, S., Demeulemeester, E., Herroelen, W., & Leus, R. (2006). The trade-off between stability and makespan in resource-constrained project scheduling. International Journal of Production Research, 44(2), 215-236. https://doi.org/10.1080/00207540500140914 Van De Vonder, S., Demeulemeester, E., & Herroelen, W. (2008). Proactive heuristic procedures for robust project scheduling: An experimental analysis. European Journal of Operational Research, 189, 723-733. https://doi.org/10.1016/j.ejor.2006.10.061 Wang, J. (2004). A fuzzy robust scheduling approach for product development projects. European Journal of Operational Research, 152(1), 180-194. https://doi.org/10.1016/S0377-2217(02)00701-4 Wang, L., Huang, H., & Ke, H. (2015). Chance-Constrained Model for RCPSP with Uncertain Durations. Journal of Uncertainty Analysis and Applications, 3(1), 12. https://doi.org/10.1186/s40467-015-0034-8 Wang, X., & Huang, W. (2010). Fuzzy resource-constrained project scheduling problem for software development. Wuhan University Journal of Natural Sciences, 15(1), 25-30. https://doi.org/10.1007/s11859-010-0106-z Wang, X., & Ning, Y. (2017). Uncertain chance-constrained programming model for project scheduling problem. Journal of the Operational Research Society. https://doi.org/10.1057/s41274-016-0122-2 Xu, J., & Feng, C. (2014). Multimode Resource-Constrained Multiple Project Scheduling Problem under Fuzzy Random Environment and Its Application to a Large Scale Hydropower Construction Project. The Scientific World Journal, 1-20. https://doi.org/10.1155/2014/463692 Yaghoubi, S., Noori, S., & Azaron, A. (2015). The Markovian Multi-Criteria Multi-Project Resource-Constrained Project Scheduling Problem. In Handbook on Project Management and Scheduling, 2, 837-862. https://doi.org/10.1007/978-3-319-05915-0 Yang, L., Fu, Y., Li, S., Huang, B., & Tao, P. (2008). A buffer sizing approach in critical chain scheduling with attributes dependent. In International Conference on Wireless Communications, Networking and Mobile Computing, WiCOM, 1-4. https://doi.org/10.1109/WiCom.2008.1805 Zafra-Cabeza, A., Ridao, M.A., & Camacho, E.F. (2004). Chance constrained project scheduling under risk. Conference Proceedings - IEEE International Conference on Systems, Man and Cybernetics, 2, 1789-1794. https://doi.org/10.1109/ICSMC.2004.1399903 Zammori, F.A., Braglia, M., & Frosolini, M. (2009). A fuzzy multi-criteria approach for critical path definition. International Journal of Project Management, 27(3), 278-291. https://doi.org/10.1016/j.ijproman.2008.03.006 Zhang, J. (2015). A bi-objective Model for Robust Resourceconstrained Project Scheduling Problem with Random Activity Durations. In Proceedings of 2015 IEEE 12th International Conference on Networking, Sensing and Control, 28-32. Zhang, J., Song, X., & Díaz, E. (2016). Project buffer sizing of a critical chain based on comprehensive resource tightness. European Journal of Operational Research, 248(1), 174-182. https://doi.org/10.1016/j.ejor.2015.07.009 Zhang, J., Song, X., & Díaz, E. (2017). Critical chain project buffer sizing based on resource constraints. International Journal of Production Research, 55(3), 671-683. https://doi.org/10.1080/00207543.2016.1200151 Zhang, M., & Chen, R. (2008). Buffer Sized Technique in Critical Chain Management: A Fuzzy Approach. In 4th International Conference on Wireless Communications, Networking and Mobile Computing, 7393-7396. Zhang, X., Cui, N., Bie, L., & Chai, Y. (2011). Timely project completion probability and stability cost on the interaction among uncertainty of random duration, service level and feeding buffer in a RCPSP environment. In 2011 International Conference on Management and Service Science, 2-5. -133- Journal of Industrial Engineering and Management - https://doi.org/10.3926/jiem.2492 Zhang, Z. (2014). A MODM Bi-level Model with Fuzzy Random Coefficients for Resource-Constrained Project Scheduling Problems. In Seventh International Joint Conference on Computational Sciences and Optimization, 666-669. https://doi.org/10.1109/CSO.2014.123 Zhang, Z., Liu, M., Zhou, X., & Chen, L. (2015). A multi-objective DCP model for bi-level resource-constrained project scheduling problems in grounding grid system project under hybrid uncertainty. KSCE Journal of Civil Engineering, 20, 1631-1641. https://doi.org/10.1007/s12205-015-0615-6 Zhang, Z., & Xu, J. (2015). Bi-level multiple mode resource-constrained project scheduling problems under hybrid uncertainty. Journal of Industrial and Management Optimization, 12(2), 565-593. https://doi.org/10.3934/jimo.2016.12.565 Zhao, Z.Y., You, W.Y., & Zuo, J. (2010). Application of Innovative Critical Chain Method for Project Planning and Control under Resource Constraints. Journal of Construction Engineering and Management, 136(September), 1056-1060. Journal of Industrial Engineering and Management, 2018 (www.jiem.org) Article’s contents are provided on an Attribution-Non Commercial 4.0 Creative commons International License. Readers are allowed to copy, distribute and communicate article’s contents, provided the author’s and Journal of Industrial Engineering and Management’s names are included. It must not be used for commercial purposes. To see the complete license contents, please visit https://creativecommons.org/licenses/by-nc/4.0/. -134-