Brief Introduction to CCPM

Transcription

1 CRITICAL CHAIN BUFFER SIZING: A COMPARATIVE STUDY ALEXANDRA B. TENERA Department of Industrial and Mechanical Engineering University Nova of Lisboa, Caparica, Portugal Brief Introduction to CCPM activities have some uncertainty, particularly regarding their durations. The recognition that uncertainty lies at the heart of project planning has been inducing several efforts in project scheduling under uncertainty since the end of fifties (Ang, Abdelnour, & Chaker, 1975; Diaz & Hadipriono, 1993; Herroelen & Leus, 2005). However, studies show that achieving scope, cost, or time still doesn t always happen especially in the project duration (Interlink Consulting, 2006; Tenera, 2006). Several sources of uncertainty in a project can be easily identified inducing different types of uncertainty and dealing with that may require different management approaches and strategies (De Meyer & Loch, 2002; Pich, Loch, & Meyer, 2002). The presence of uncertainty can stimulate the creation of buffers to reduce the consequences of possible disruptions. The use of project time buffers explicitly introduced in the project schedule has long been suggested (Hamburger, 1987), but more recently Goldratt (1997) promoted the intentional introduction of both project and feeding buffers in the project schedule to improve critical chain stability and to increase the probability of achieving a promised or agreed-upon project duration. Focused buffer consumption to control and to manage the project schedule execution through project uncertainty has also been promoted. This approach, best known as critical chain project management (CCPM), could be viewed as a way of trade-off between stability and makespan (Herroelen & Leus, 2005), and has been considered a promising methodology for project management (Herroelen & Leus, 2001; McKay & Morton, 1998). So, how these buffers are created and managed might be at the core of the improvement of project management. Developed using the Goldratt s theory of constraints (TOC), CCPM makes some strategic changes in how projects are traditionally planned, scheduled, and managed: CCPM emphasizes resource behavior in project schedule aiming to reduce project schedule changes and time overruns. The critical chain project planning and scheduling addresses uncertainty in form of activity duration variation. Premeditated introduction of time buffers in project schedule along with buffer consumption is used to manage the project schedule; To reduce project delays caused by Parkinson s Law (work expands to fill all the time available) but at the same time, accounting for Murphy s Law critical chain scheduling (Patrick, 1999) promotes several mechanisms: Build the schedule with target durations that are too tight to allow diversion of attention; Don t use activities due dates; Don t allow multitasking and resources interruptions. In CCPM the critical chain is the set of activities that defines the overall duration of the project, taking into account precedence and also resource dependencies. Critical chain requires the construction of a feasible schedule with only the time to do the work in each activity. This estimate is usually described as corresponding to a 50% confidence level or an average duration. Resource conflicts are resolved by moving tasks earlier in time (Newbold, 1998). There are two general kinds of resources in CCPM projects: the ones who perform critical activities and those who don t. It is necessary to make ensure that critical chain resources are available when the preceding activity is done, without relying on fixed due dates. For that resource buffers are considered in schedule, which traditionally works as wake-up calls in schedule. If one activity is completed before its scheduled duration the work done should be immediately forward, this behavior called relay runner tries to make use of all the occurred anticipations. 1

2 One of the main steps of CCPM to build the project schedule is to pull out the safety of each estimated activity duration and get target duration for each activity. Then, in the following steps, is to aggregate all of the safety at the end of the project, where it acts as a protection of the project due date, called project buffer (PB). Feeding buffers (FB) are placed whenever a non-critical activity joins the critical chain, in order to protect the critical activities against negative variations of the feeding chains. Like in the TOC applied to production, buffer management is the main schedule control mechanism used, which needs to assess buffer penetration or consumption in order to accepted rescheduling or not. As an activity takes longer than the schedule foresees, buffers are consumed. On the other hand, as they take less time, those buffers are replenished. Therefore, one of the main challenges of the CCPM is the adequate sizing of the time buffers. If the buffers were dimensioned beyond the necessary size, immediate practical implications happen, for example: unnecessary addition of costs and/or anticipated investments or eventual losses of market chances. On the other hand, if buffers were underestimated it may increase the probability of duration overruns, which can represent financial penalties and reliable loss from the part of the customer or market. If buffers are adequately sized, the project conclusion date should be satisfied and rarely exceeded. However, in both cases the size of the project buffer can depend on the desired probability for completing the project schedule and should be determined according to it. After the Goldratt s (1997) proposal of the critical chain principles of scheduling and management for projects, several authors suggested different methods for sizing projects and feeding buffers, which will be presented and discussed in the following section. Current Buffer Sizing Methods CCPM builds schedules with activities target durations tight and finished to start (FS) dependencies in a schedule as late as possible (ALAP). A safety margin is aggregated at the end of the project, where it acts as a protection of the project due date, called project buffers (PB), and in feeding buffers (FB), placed whenever a non-critical activity joins the critical chain, to protect critical activities against variations of the feeding chains. For project and feeding buffers sizing (BfS) several methods have been proposed. Goldratt (1997) suggested a practical and simple cut of 50% in the pessimistic duration (d Ci ) and to schedule a buffer of 50% of the trimmed duration of the chain with n activities. Variations of the method have been used considering 50% of the sum of the differences between a tendency measure (T) and the estimated pessimistic duration of the chain activities (n) as presented in formula (1), which permits considering the asymmetry of the underlying variability in activity. n C (1) i BfS = 0.5 ( d T ) 11 = Newbold (1998) and Leach (2000) detailed and developed the Goldratt s critical chain concepts and launch two other classical methods. Newbold (1998) revealed the approximated formula (2) assuming lognormal distributions functions for activity duration with mean μ ˆ i and pessimistic durations, d Ci. n i= 1 ( d ˆ C i) 2 BfS = 2 μ /2 (2) i This approximated formula was developed assuming: lognormal distributions functions for activity duration, pessimistic durations, d Ci, associated to a 90% confidence level on the the estimation, the applicability of the Central Limit Theorem and two standard deviations for sizing the buffer. Herroelen and Leus (2001) showed that in these conditions the difference considering 90% durations and the estimated mean durations, μ ˆ i, should have a smaller number of standard deviation then the two considered, which could result in practical terms in smaller buffers than expected if the assumed conditions occurs. Leach (2000) considered the square root of the sum of squares of the differences between the upper bounds of the estimated activity duration, U i, and the lower bounds, L i, for all activities of the longest chain feeding the buffer (3), 2

3 assuming no specific distribution function. This formula can be considered a generalization of the formula (2); it is a distribution free formula, and; if the lower bound is considered to be the mean duration and the upper bound the pessimistic duration considered on (3) both formulas will produce the same results. n i= 1 ( ) 2 i (3) i BfS = U L Other less known methods have also been proposed. Hoel and Taylor (1999) suggested that the size of the project buffer should depend on the desired probability for completing the project schedule and would be determined according to it. They used Monte Carlo simulation to project buffer sizing considering the difference between simulated percentile duration and the simulated mean and no feeding buffers. In this situation if the schedule slack is less than the expected variability of the non critical chains, ignoring feeding buffers can produce impacts on critical activities and this will be against the theoretical principles of the critical chain scheduling (Goldratt, 1997). Shou and Yeo (2000) proposed to estimate project buffers sizes by classifying activities in four different uncertainty levels according to the activities relative dispersion and the level of safety desired. They suggested that the same method could be applied to feeding buffers but they didn t try it. Park and Penã-Mora (2004) proposed applying simulation and system dynamics to buffer sizing in specific projects such as fast track construction and concurrent design projects (2006). In these last approaches buffers sizes and positions are continuous updated according to the dynamics of project execution in these special types of projects. The SMC Approach The need for practical and proper sizing of time buffers in CCPM scheduling led to the development of a different approach to buffer sizing based on the CCPM concepts and by using a simulation technique (Monte Carlo) the SMC method (which stands for Simulação para a Melhoria da Calendarização). The underlying logic of the proposed method is to size buffers addressing the risk of overrun the scheduled date of last activities of the chains (Tenera, 2006), the buffers will be designed considering the project schedule in the ALAP logic, as proposed in CCPM methodology. Considering, as an example, a very simple project network without resource constraints (Figure 1) for simplification with activities, A i, interrelated with FS activities relations, schedule in an ALAP logic assuming target durations for each activity. From Figure 1 it is possible to identify activities A1, A2, A3 e A4 as the critical activities of the project and activities A5 and A6, as non critical activities. A1 SNET A2 A3 A4 PB dc Shed dc P A5 SNET A6 FB dc Shed dc P Figure 1. SMC Logic 3

4 Assuming the ALAP scheduling logic of CCPM, the start of non critical activities, not depending from preceding activities conclusion, should not start before their ALAP schedule start date (with buffers included). Time buffers should be sized using: an assumed risk level for time overrun of the last activity of the critical chain for project buffer sizing purpose (PB), and a risk level for time overrun of end schedule dates of non critical activities converging to critical activities in sizing feeding buffers (FB). To consider the effect of the ALAP schedule in the activities whose start schedule are not depending on precedence activities conclusion, start no earlier than (SNET) constraints can be imposed on the ALAP schedule while others should have a start ASAP, according to the relay runner behavior execution of the CCPM methodology. This logic can be called as an as soon as needed (ASAN) logic. Using this ASAN logic and network simulation the conclusion of the last activity of which chain converting to the critical chain and the last activity of the critical chain (A6 and A4 in the example) can be evaluate and the risk of overrun the schedule conclusion dates for these activities characterized. To define the risk of overrun schedule activities end time, Monte Carlo Simulation was chosen as the simulation tool; it is a well known tool, and it has been used in project management risk analysis. The use of Monte Carlo Simulation as a tool for project buffer sizing was already proposed by Hoel and Taylor (1999), but in their proposal the ALAP scheduling logic, as well as the use of explicit feeding buffers appeared not to be considered. The difference between the simulated end date of these activities, associated with a chosen confidence level (dc p ), and the schedule date (dc Shed ), will be the time buffer to schedule (4). This buffer is intended to reduce the impact of non critical activities in the critical chain and to protect the project end date against variability. Regarding this, feeding buffers insertion can require the creation of a gap on the critical chain, as the duration variability of non critical chain should not delay the schedule start of critical chain activities (Goldratt, 1997) in order to improve the stability of the critical chain. BfS = dc dc (4) p Shed Experimental Design To assess the impact of sizing methods, considering different network sizes, a block design logic approach was used (see Figure 2). The effect of the sizing method on the schedule was considered the main study variable although it was assumed that others factors, such as network size (block), could also affect the results. These two factors could have i levels and j networks. Each network had different network complexity (NC) and random uncertain duration (defined by a tendency variable (T), an optimistic duration (d Cg ) and pessimistic (d Ci ) duration). Using ANOVA analysis and statistical tests we could evaluate the impact of different buffer sizing on the network. Block Nets Nets. Nets Factor 1 2 j Method 1 (1) (2). (k) Method 2 (1) (2). (k) Method i (1) (2). (k) (1) = f 1 (NC, T, d Ci, d Cg ) (2) = f 2 (NC, T, d Ci, d Cg ) (k) = f k (NC, T, d Ci, d Cg ) Figure 2. Experimental Design Approach Several Resource Constrained Scheduling Problems (RCPSP) datasets are available to researchers in the Scheduling Problem Library (PSPLIB) on the internet ( To address the effect of buffer sizing methods, samples of networks, generated by ProGen for the Single Mode RCPSP (SMRCPSP), were used because: CCPM scheduling can be integrated in the resource constrained project scheduling problems; CCPM advocates that projects can be well modeled by a network where activities are represented by nodes, FS activity interdependency and all activities must be executed. Although other network sets have been used in other 4

5 works (Herroelen & Leus, 2001) ProGen data sets are generated by controlled design parameters (Kolisch, Sprecher, & Drexl, 1995). To model uncertainty in activity duration, Triangular distributions were used. These distributions are particularly interesting for risk simulation when little data is available (Law & Kelton, 1991; Vose, 1996); they present good results for risk analysis (Johnson, 2002); they are considered conceptually easy to communicate and implement and they admit several types of asymmetry. In this study, distributions were mostly right skewed (activities with durations within [m/3, 2m] and [2/3m, 2m] limits), but left skewed (activities with durations within [1/3m, 4/3m] limits) or symmetric (activities with durations within [1/3m, 4/3m] limits) were also included. The variable m represented three times the original activity deterministic duration and the optimistic and pessimistic limits were randomly generated in Excel. For sizing the buffers, three methods were considered: A variant of Goldratt s method based on formula (1), where the variability removed of each activity duration was the difference between the pessimistic estimated duration (the upper limit) and the modal value (m). The results are associated with the M50 designation. As proposed by Leach (2000) and designated by SQR, formula (3) was used with U i equals to the pessimistic estimated duration and L i the modal value (m); The proposed method, SMC, considering the difference of 95% confidence level on the simulated final date and the schedule final date as presented in Figure 1. The critical chain and feeding chains identification and the scheduling of each project were done with ProChain, which was used to calculate the buffers in the case of the M50 and SQR. In the SMC case the buffers were simulated using 1000 iterations software after RiskNet and Excel conversion of networks to Microsoft. To evaluate and compare the impact of the three methods, each project schedule was again simulated under the SNET and ASAP conditions Two main groups of measures were used to evaluate the impact of the CC/BM scheduling logic and the considered buffer sizing methods namely morphological and time measures. In the comparative evaluation of the impact four main aspects were assessed: buffers morphological impact, buffers sizes impact, buffers impact on project schedule duration and schedule lateness (Tenera, 2006) as described next. Morphological Impact Measures The following indicators measure the impact of CC/BM scheduling logic: Number of feeding buffers by the number of activities in the critical chain, n(fb s) / n(cc); Number of feeding buffers by the number of activities of the deterministic project schedule, n(fb s) / N; Number of activities in the critical chain by the number of activities of the deterministic project schedule, n(cc) / N. The ratios n(fb s) / n(cc) and n(fb s) / N are indicators of control points density during project execution giving a measure of control effort of CC/BM logic relating to the critical chain dimension and to project dimension. The n(cc) / N, is an indicator of the capacity of focusing of the CC/BM scheduling logic. Time Impact Measures To measure the impact of the sizing methods in the deterministic project schedule were considered several indicators to characterize three different aspects: Buffers sizes: on the project buffer (PBD) and feeding buffers (FBMD) Impact on project schedule duration: considering only a project buffer (DPB), with project and feeding buffers inserted (DB) and the DB / DD measuring the impact in schedule of the assumed uncertainty in project activities during project execution 5

6 Usage of buffers inserted on schedule: project buffer consumption (PBC) and feeding buffers consumption (FBMC), percentage of buffers inserted but not used (FBNC) and the standardized schedule lateness (ML/DD) The main results obtained are presented and discussed in the following section. Buffers Morphological Impact In Figure 3 the number of the network critical activities (n(cc)) by the number of the network activities (N) for all used networks represented by the number of the activities (30, 60, 120) and a configuration parameter, P, varying from 1 to 5, are presented. From Figure 3 it can be seen that although the number of activities in the project doubled the number of critical activities didn t increased so much representing only 9% to 30% of the networks activities. It should be noted that this proportion is higher for smaller projects than for projects with more activities. 0,40 0,30 n (CC) / N 0,20 0,10 0, Combination of the configuration parameters - P 30P 60P 120P Figure 3. Networks Critical Activities Evaluating the impact of the feeding buffers introduced in schedule (n(fb s)) and the number of the network activities (N) ( Figure 4) we can see that the number of feeding buffers incremented the network nodes between 9% and 30%. These especial nodes represent in CCPM scheduling control points of the project time performance and once again they were proportionally more in smaller projects than in bigger projects. 6

7 0,40 0,30 n (FB's) / N 0,20 0,10 0, Combination of the configuration parameters - P 30P 60P 120P Figure 4. Feeding Buffers Weight in Networks Because the ratios n(cc) / N and n(fb s) / N could be considered as a measure of CC/BM focusing and control effort, results indicate that CCPM scheduling should be easier to implement in bigger projects than in smaller ones, because big projects allow a more focused management than in a smaller one. In the following sections for graphical identification of the networks the digit 3 will be used to represent networks with 30 activities, 6 to designate networks with 60 activities and the number 12 stands for networks with 120 activities. Last digit represents the configuration parameter, P, varying from 1 to 5. Buffers Sizes For sizing buffer evaluation PB duration (PBD) and the mean of the feeding s buffers durations (FBMD) were characterized. Comparing the project buffers of the three methods it can be found in the analysis that SMC generally generates smaller project buffers than the M50 and SQR and the M50 produces the largest ones (see Figure 5). Identical results can be seen in Figure 6 for feeding buffers mean sizes and in this case SMC detach more from the other methods. It should be noted that in smaller networks (with 30 and 60 activities) the SQR method produced on average higher feeding buffers than M50. 7

8 PBD [days] M50 SQR SMC Figure 5. Buffer Sizes FBMD [days] M50 SQR SMC Figure 6. Mean Feeding Buffers Sizes Buffers Impact on Scheduled Duration The results of the impact of buffers insertion, in terms of time, on deterministic schedule, are represented in Figure 7. As expected, initial deterministic duration of the project, using modal duration without buffers (DD), increased 8

9 when all buffers were inserted (DB) for all evaluated methods. The increase in the project duration revealed to be rather irregular (from 10% to 80%). Figure 7 shows that SMC method reduces the schedule duration of the projects considering project and feeding buffers schedule (DB), except in one case. This reduction was a consequence of smaller feedings and project buffers obtained with this method (Tenera & Cruz-Machado, 2007). The M50 method generally generated the longest schedules. DB / DD Figure 7. Buffers Impact on the Deterministic Schedule M50 SQR SMC In order to evaluated which type of buffers produces more impact on the project schedule, the increased on the deterministic duration caused only from project buffer insertion (DPB-DD) and the increased caused only by feeding buffers insertion (DB-DPB) standardized by the deterministic duration (DD) were calculated and results are exemplified in Figure 8 and Figure 9, for M50 and SMC, respectively Increment [percentage] (DPB-DD) / DD (DB-DPB) / DD Figure 8. M50: Buffers Impact 9

10 Increment [percentage] (DPB-DD) / DD (DB-DPB) / DD Figure 9. SMC: Buffers Impact Comparing the results from Figures 8 and 9, we can see that, although exceptions can occur, feeding buffers insertion produced smaller impact on the project schedule time than project buffer. The SMC method generated less impact than M50. These results can be extended to SQR method as we can see in Figure Increment [percentage] (DPB-DD) / DD (DB-DPB) / DD Figure 10. SQR: Buffers Impact 10

11 Buffers Consumption Buffer consumption results are presented in Figure 11 representing project buffer consumption (PBC) and mean feeding buffers consumption of the buffers used (FBMC) in M50 method and the proposed method, SMC, for illustrative proposes Buffer consumption [per M50: PBC M50: FBMC SMC: PBC SMC: FBMC Figure 11. Buffers Consumption Comparing the results of both methods we see that SMC have superior buffer usage than M50, because the consumption levels for both project and feeding buffers are higher. Results also show that in all evaluated methods project buffer was never exceeded, which is the main purpose of CCPM scheduling. Even if they proved to be able to not overrun the project end date, they also had always unused project buffer and the level of the project usage was generally lower than the mean consumption of the feeding buffers used. Although not shown it should also be notice that some buffers weren t used that indicates new research should be developed to improve feeding buffers schedule insertion. The difference between schedule project due date defined by DB and the simulated project conclusion date, considering the 95 percentile, was always positive in all projects and in all methods (Tenera, 2006). The results showed that project buffer was never totally consumed. This indicates that SMC as well as the other analyzed methods is able to not overrun project conclusion date and maybe considered somehow optimistic but SMC produces better results. Schedule Lateness To evaluate the schedule lateness of each method, a mean lateness (ML) was calculated by the arithmetic mean of the differences between project due date (defined by the DB) and the simulated mean date of feedings and project buffer. The result was standardized by the deterministic duration (DD) of each project. The values obtained with SMC are lower than in the other methods (Figure 12), which means that the mean consumption of buffers are closer to project schedule due date than in other methods. So, with SMC activities are concluded later than in the methods assessed improving the CCPM ALAP logic. In general, the results show that: Generally, buffers are reduced using the proposed method, than using classical methods; This reduction allows estimated project duration increasing project competitiveness proposal; buffers generally produced more impact in project duration than feeding buffers; 11

12 SMC method showed more buffer consumption than the other methods tested also without overruns; The proposed method, SMC, allows higher lateness than other methods improving the use of the ALAP logic and its potential benefits ML / DD M50 SQR SMC Figure 12. Schedule Lateness According to the obtained results, the proposed method contributes for a better time buffer sizing, producing reduced buffers, better estimated project durations, increasing schedule lateness and higher buffer consumption without overrun the estimated project duration. Conclusions This paper reviews different methods of project and feeding buffers sizing. The research on the impact of scheduling buffers against uncertainty in activity durations in projects used 15 networks and 60 simulations and three different methods of feeding and project buffer sizing were investigated. The research results showed that buffers can reduce the risk of project duration overruns, using feeding buffers can impact the project buffer size reducing it and project buffer generally produces more impact in project duration than feeding buffers. In this study the impact of buffer management wasn t considered assuming that if they were used we could reduce the project buffer even more without project duration overruns, hypotheses that should be investigated in the future. The research also shows that some of the feeding buffers inserted weren t used; thereby the insertion of feeding buffers needs further studies. The investigation revealed that CCPM is still available for improvements and the most known methods for buffer sizing can be considered optimistic as well as the proposed method, which should be improved by using at least a double step simulation process. The first step for feeding buffer sizing and the second to project buffer sizing considering the feeding buffers already inserted. As indicated by the experimental results the use of Monte Carlo simulation can improve the estimated project conclusion and the proposed method, SMC, has as main advantages: and feeding buffers are defined considering all activities and network characteristics like activity and resource interdependencies considering a ALAP logic; Generates better and reduced project duration than in the other two methods analyzed; The method uses well known simulation technique like Monte Carlo which is conceptually simple, flexible and taught in graduation schools; Can be applied using established computer applications available for use in real projects. 12

13 Nevertheless, the used of the Monte Carlo simulation technique for buffer sizing implies that organizations have the ability to follow the schedule assumptions and the estimation process is accurate. Otherwise, like in other methods available, the used of this tool can always be questionable. Acknowledgments The authors would like to thank the financial support from FLAD and also to ProChain Solutions and AGI for technical support received during this research. References Ang, A. H.-S., Abdelnour, J., & Chaker, A. A. (1975). Analysis of activity networks under uncertainty. Journal of the Engineering Mechanics Division, 101(4), De Meyer, A., & Loch, C. H. P., Michael T. (2002). Managing project uncertainty. MIT Sloan Management Review, 43(2), Diaz, C. F., & Hadipriono, F. C. (1993). Nondeterministic networking methods. Journal of Construction Engineering and Management, 119(1), Goldratt, E. M. (1997). Critical chain. Great Barrington, MA, EUA: North River Press. Hamburger, D. H. (1987). "On time" project completion-managing the critical path. Management Journal, XVIII(4), Herroelen, W., & Leus, R. (2001). On the merits and pitfalls of critical chain scheduling. Journal of Operations Management, 19(5), Herroelen, W., & Leus, R. (2005). scheduling under uncertainty: Survey of research potentials. European Journal of Operation Research, 165(2), Hoel, K., & Taylor, S. G. (1999). Quantifying buffers for project schedules. Production and Inventory Management Journal, 40(2), Interlink Consulting. (2006). Organizational project management baseline study. Retrieved November, 14, Johnson, D. (2002). Triangular approximations for continuous random variables in risk analysis. Journal of the Operational Research Society, 53(4), Kolisch, R., Sprecher, A., & Drexl, A. (1995). Characterization and generation of a general class of resourceconstrained project scheduling problems. Management Science, 41(10), Law, A. M., & Kelton, D. (1991). Simulation modeling and analysis (2ª ed.). Singapura: McGraw-Hill. Leach, L. (2000). Critical chain project management. Norwood, MA, EUA: Artech House. Lee, S., Park, M., & Pena-Mora, F. (2006). Reliability and stability buffering approach: Focusing on the issues of errors and changes in concurrent design and construction projects. Journal of Construction Engineering and Management, 132(5), McKay, K. N., & Morton, T. E. (Artist). (1998). Critical chain [Critical Chain]. Newbold, R. C. (1998). management in the fast lane. Boca Raton, FL, EUA: St. Lucie Press. Park, M., & Pena-Mora, F. (2004). Reliability buffering for construction projects. Journal of Construction Engineering and Management, 130(5), Patrick, F. S. (1999, April). Getting out from between Parkinson's rock and Murphy's hard place. PM Network, 13, Pich, M. T., Loch, C. H., & Meyer, A. (2002). On uncertainty, ambiguity, and complexity in project management. Management Science, 48(8),

14 Shou, Y., & Yeo, K. T. (2000, November 12-15). Estimation of project buffers in critical chain project management. Paper presented at the IEEE International Conference on Management of Innovation and Technology, Singapore. Tenera, A. (2006). Contribution for the improvement of uncertainty management in project duration through theory of constraints. Unpublished PhD Thesis, Universidade Nova de Lisboa, Lisboa. Tenera, A., & Cruz-Machado, V. (2007). Critical chain project management: A new approach for time buffer sizing. Paper presented at the Institute of Industrial Engineering Annual Conference, Nashville. Vose, D. (1996). Quantitative risk analysis: A guide to Monte Carlo simulation modelling. Chichester, RU: John Wiley & Sons. 14

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