The vast majority of the research efforts in project risk management tend to assess cost risk and schedule risk independently. However, project cost and time are related in reality and the relationship between them should be analyzed directly. We propose an integrated cost and schedule risk assessment model for complex product systems R&D projects. Graphical evaluation review technique (GERT), Monte Carlo simulation, and probability distribution theory are utilized to establish the model. In addition, statistical analysis and regression analysis techniques are employed to analyze simulation outputs. Finally, a complex product systems R&D project as an example is modeled by the proposed approach and the simulation outputs are analyzed to illustrate the effectiveness of the risk assessment model. It seems that integrating cost and schedule risk assessment can provide more reliable risk estimation results.
Complex product systems can be defined as high cost, engineering-intensive products, systems, networks, and constructs [
The traditional approach to estimating risk of cost and schedule has been to estimate them independently (e.g., [
In recent years, many approaches have been proposed to carry out relative research into integrated analysis problems of stochastic cost and schedule. Carr [
Xu et al. [
In complex product systems R&D project, rework of an activity is common, which is usually caused by probabilistic failure to meet the planned design objective. GERT considered such failure probability of activities and could be used to model and analyze the process of complex product systems with Monte Carlo simulation [
In the paper, motivated by the successful use of the GERT, we employ this technique with multifeedback branches to describe the process of complex systems R&D project. On the basis of the theory of probability and the relationship of marginal probability distribution function, conditional probability distribution function, and integrated probability distribution function, we construct an integrated cost and schedule risk estimation model. Finally, an example of a real-life project is analyzed with the integrated risk estimation model in detail. Compared with the prediction estimation using approximate curve surface, our method can estimate risk probability more rapidly and accurately by processing Monte Carlo simulation results.
The remainder of this paper is organized as follows. Section
Suppose that the simulation runs
Let
The integrated cost and schedule risk is defined as the probability of failing to complete the project under a specified objective with respect to cost and schedule. The integrated cost and schedule frequency statistics is shown in Figure
The integrated cost and schedule frequency statistics.
Let
The integrated probability distribution function in region
Accordingly, the integrated cost and schedule risk probability distribution function is
The conditional probability density functions of cost and schedule are
Our integrated cost and schedule risk estimation model is demonstrated with example data from an obstacle clearance armored vehicle (OCAV) R&D project. We utilize GERT to model the OCAV R&D process, which is usually divided into five phases: project feasibility study, project evaluation, subsystems development, system assembly, and integrated testing. The OCAV is composed of six subsystems. To focus on the simulation-based method presented in this paper, we name the six subsystems as subsystems 1 to 6 instead of their actual names.
Figure
Distribution of time and cost for the activities.
Activities | Number | Time (month) | Cost (million) | ||||
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Project feasibility studies |
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6 | 9 | 12 | 0.50 | 1.00 | 1.50 |
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Program demonstration |
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3 | 6 | 9 | 6.00 | 6.50 | 7.00 |
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Subsystem 1 development |
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2 | 3 | 4 | 0.20 | 0.30 | 0.40 |
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12 | 14 | 16 | 9.00 | 11.00 | 13.00 | |
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10 | 12 | 12 | 1.80 | 2.00 | 2.20 | |
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Subsystem 2 development |
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1 | 2 | 3 | 0.05 | 0.10 | 0.15 |
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10 | 12 | 14 | 6.00 | 8.00 | 10.00 | |
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10 | 12 | 12 | 2.00 | 3.00 | 4.00 | |
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Subsystem 3 development |
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2 | 3 | 4 | 0.15 | 0.20 | 0.25 |
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10 | 12 | 14 | 5.00 | 7.00 | 9.00 | |
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10 | 12 | 12 | 2.00 | 3.00 | 4.00 | |
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Subsystem 4 development |
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1 | 2 | 3 | 0.05 | 0.10 | 0.15 |
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8 | 10 | 12 | 1.30 | 1.50 | 1.70 | |
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6 | 10 | 12 | 0.20 | 0.30 | 0.40 | |
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Subsystem 5 development |
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1 | 2 | 3 | 0.05 | 0.10 | 0.15 |
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6 | 8 | 10 | 0.90 | 1.00 | 1.00 | |
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8 | 10 | 12 | 0.10 | 0.20 | 0.30 | |
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Subsystem 6 development |
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1 | 2 | 3 | 0.05 | 0.10 | 0.10 |
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10 | 12 | 14 | 0.60 | 0.80 | 1.00 | |
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10 | 12 | 12 | 0.10 | 0.20 | 0.30 | |
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System assembly |
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2 | 4 | 6 | 0.30 | 0.40 | 0.50 |
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Integrated testing |
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9 | 12 | 15 | 800 | 1000 | 1200 |
Stochastic multifeedback GERT model of the OCAV R&D project.
Simulation runs
The integrated and marginal frequency distribution of total cost and duration.
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53 |
56 |
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74 |
77 |
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104 |
107 |
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0 | 0 | 0 |
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0 | 0 |
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0 | 0 |
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5300 |
0 | 1 | 0 |
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0 | 0 |
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0 | 0 |
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5600 |
0 | 0 | 0 |
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4 | 1 |
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0 | 0 |
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7400 |
0 | 0 | 0 |
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10 | 17 |
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0 | 0 |
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7700 |
0 | 0 | 0 |
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5 | 11 |
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2 | 0 |
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10400 |
0 | 0 | 0 |
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0 | 0 |
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0 | 0 |
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10700 |
0 | 0 | 0 |
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0 | 0 |
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0 | 0 |
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We can also obtain the following results. According to the simulation results, the mean and standard deviations of the total cost and duration are calculated as follows:
Frequency histograms of total cost and duration can be drawn and are shown in Figures We perform the goodness of fit test and deduce that the probability distribution of the total cost and duration is normal distribution. More specifically, the theoretical probability distribution functions of the total cost and duration are subject to From the result in (3), we believe that the total cost and duration are all subject to normal distribution. Hence, interval estimation of parameter can be carried out. The confidence interval of 95% for the total cost The confidence interval of 95% for the total duration
The frequency histograms and probability distribution of total cost.
The frequency histograms and probability distribution of total duration.
We can get the marginal probability density functions of the total cost and total duration
Marginal risk probability distribution of total cost and duration.
Total cost |
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Total duration |
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53 | 0.001 | 0.001 | 0.999 | 53 | 0.000 | 0.000 | 1.000 |
56 | 0.005 | 0.006 | 0.994 | 56 | 0.001 | 0.001 | 0.999 |
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77 | 0.109 | 0.593 | 0.407 | 77 | 0.086 | 0.225 | 0.775 |
80 | 0.122 | 0.715 | 0.285 | 80 | 0.130 | 0.355 | 0.645 |
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107 | 0.003 | 0.999 | 0.001 | 107 | 0.001 | 1.000 | 0.000 |
110 | 0.001 | 1.000 | 0.000 | 110 | 0.000 | 1.000 | 0.000 |
The marginal probability distribution and the marginal risk probability distribution of the total cost and the total duration are shown in Figures
The marginal probability distribution and marginal risk probability distribution of total cost.
The marginal probability distribution and marginal risk probability distribution of total duration.
According to formulas (
Integrated probability distribution of total cost and duration
Total cost |
Total duration | |||||||
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53 | 56 |
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74 | 77 |
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107 | 110 | |
5300 | 0.000 | 0.000 |
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0.000 | 0.000 |
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0.001 | 0.001 |
5600 | 0.000 | 0.001 |
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0.003 | 0.003 |
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0.006 | 0.006 |
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7700 | 0.000 | 0.001 |
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0.135 | 0.210 |
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0.593 | 0.593 |
8000 | 0.000 | 0.001 |
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0.139 | 0.219 |
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0.715 | 0.715 |
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10700 | 0.000 | 0.001 |
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0.139 | 0.225 |
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0.999 | 0.999 |
11000 | 0.000 | 0.001 |
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0.139 | 0.225 |
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1.000 | 1.000 |
Integrated risk probability distribution of total cost and duration
Total cost |
Total duration | |||||||
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53 | 56 |
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74 | 77 |
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107 | 110 | |
5300 | 1.000 | 1.000 |
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1.000 | 1.000 |
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0.999 | 0.999 |
5600 | 1.000 | 0.999 |
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0.997 | 0.997 |
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0.994 | 0.994 |
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7700 | 1.000 | 0.999 |
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0.407 | 0.407 |
8000 | 1.000 | 0.999 |
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0.285 | 0.285 |
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10700 | 1.000 | 0.999 |
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0.861 | 0.775 |
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0.001 | 0.001 |
11000 | 1.000 | 0.999 |
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0.861 | 0.775 |
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0.000 | 0.000 |
According to the data in Table
Curved surface of integrated risk probability distribution of total cost and duration.
According to formula (
Conditional probability distribution of the total cost
Total cost |
Total duration | |||||||
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53 | 56 |
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74 | 77 |
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107 | 110 | |
5300 | 0.000 | 0.000 |
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0.000 | 0.000 |
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0.001 | 0.001 |
5600 | 0.000 | 1.000 |
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0.022 | 0.013 |
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0.006 | 0.006 |
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7700 | 0.000 | 1.000 |
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0.971 | 0.933 |
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0.593 | 0.593 |
8000 | 0.000 | 1.000 |
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1.000 | 0.973 |
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0.715 | 0.715 |
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10700 | 0.000 | 1.000 |
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1.000 | 1.000 |
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0.999 | 0.999 |
11000 | 0.000 | 1.000 |
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1.000 | 1.000 |
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1.000 | 1.000 |
Conditional probability distribution of the total duration
Total cost |
Total duration | |||||||
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53 | 56 |
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74 | 77 |
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107 | 110 | |
5300 | 0.000 | 0.000 |
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0.000 | 0.000 |
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0.000 | 0.000 |
5600 | 0.000 | 0.167 |
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0.002 | 0.002 |
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0.001 | 0.001 |
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7700 | 0.000 | 0.500 |
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0.388 | 0.354 |
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0.139 | 0.139 |
8000 | 0.000 | 0.500 |
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0.556 | 0.519 |
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0.225 | 0.225 |
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10700 | 1.000 | 1.000 |
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1.000 | 1.000 |
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1.000 | 1.000 |
11000 | 1.000 | 1.000 |
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1.000 | 1.000 |
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1.000 | 1.000 |
According to the above results, the smooth curves of conditional probability distribution of the total cost and the total duration are drawn in Figures
Conditional probability distribution
Conditional probability distribution
Using goodness of fit test method, we infer that the conditional probability distributions of the total cost and the total duration follow normal distributions, and the goodness of fit is good. Meanwhile, the mean
Mean and standard deviation of conditional probability distribution of total cost and duration.
Conditional distribution of total duration | Mean estimation |
Standard deviation estimation |
Conditional distribution of total cost | Mean estimation |
Standard deviation estimation |
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73.00 | 11.41 |
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5450.00 | 0.00 |
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73.06 | 6.47 |
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5450.00 | 0.00 |
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74.81 | 6.66 |
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5450.00 | 0.00 |
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81.12 | 7.93 |
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6900.14 | 705.49 |
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83.25 | 8.24 |
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7476.43 | 929.64 |
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83.28 | 8.25 |
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7477.40 | 929.68 |
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83.30 | 8.26 |
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7477.40 | 929.68 |
Through the data in Table
After that, the risk probability value
Following the definition of the conditional probability distribution function and the marginal probability distribution function, the integrated risk probability of the total cost and duration is calculated as follows:
For example, we also get the risk probability value With the data from Table Calculate the conditional probability distribution values by normal distribution:
According to formula ( Through formula (
By analyzing the output data of 1000 times simulation, we can calculate the frequency of occurrence that the total cost
For purposes of comparison, we also report results based on the curved surface method of integrated risk probability. The prediction result is
The above comparison results of relative error reveal that both of the methods are practical. However, it is too complicated to build the equation when using the approximate curve surface method. And the method proposed in this paper can estimate risk probability more rapidly and accurately by processing one set of Monte Carlo simulation results.
Based on GERT multifeedback simulation and the theory of probability distribution, we have presented an integrated risk model of cost and schedule. Using statistical analysis and Monte Carlo techniques, we get the marginal probability distribution functions and the conditional probability distribution functions. Finally, the integrated probability distribution functions of cost and schedule and risk probability distribution function are obtained. This method is proven to be more accurate. However, we know that there are many other factors in complex product systems except cost and schedule, such as environment and project resources. In the future, we will study the risk assessment about more elements.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the Innovation Foundation of BUAA for Ph.D. Graduates and the National Science Foundation of China under Grant 71271019.