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1、Hierarchy probability cost analysis model incorporate MAIMS principle for EPC project cost estimation4. Hierarchy integrated probability cost analysis (HIPCA) models for EPC cost estimation.In this section we introduce hierarchy probability cost analysis (HIPCA) methodology, which incorporates all a

2、forementioned concepts for determining the total project cost (TPC) of EPC projects. Our objective is to develop an optimal but realistic TPC for a given probability of success (PoS) that we assume has been specified by allocating the baseline budgets, and managing contingency, based on the desire t

3、o win the project and risk tolerance.4.1. Correlation coefficient and its feasible verificationOnce historical data is available, two different measures are used to reflect the degree of relation between cost elements in literature. The first one is an ordinary product-moment (Pearson) correlation c

4、oefficient and the second is a rank (Spearman) correlation coefficient. A non-parametric (distribution-free) rank statistic proposed by Spearman in 1904 as a measure of the strength of the associations between two variables ( Lehmann & DAbrera, 1998). The Spearman rank correlation coefficient can be

5、 used to give a real estimate, and is a measure of monotone association that is used when the distribution of the data make Pearsons correlation coefficient undesirable or misleading.While it may be difficult to justify use of a specific numeric value to represent the correlation between two cost el

6、ements, it is important to avoid the temptation to omit the correlation altogether when a precise value for it cannot be established. Such an omission will set the correlation in question to the exact value of zero; whereas positive values of the correlation coefficient tend to widen the total-cost

7、probability distribution and thus increase the gap between a specific cost percentile (e.g., 70%) and the best-estimate cost. That is to say, the contingency could be larger. Therefore, using reasonable non-zero values, such as 0.2 or 0.3, generally leads to a more realistic representation of total-

8、cost uncertainty.Subjective judgment also finds application in specifying the cor-relations between cost elements qualitatively. To this respect, researchers can subjectively choose two groups of correlations to assess strong, moderate, and weak relations: 0.8,0.45,0.15 ( Touran, 1993) and 0.85,0.55

9、,0.25 ( Chau, 1995). Other more recent scholars explain, simply, as a rule of thumb, we can saythat correlations of less than 0.30 indicate little if any relationship between the variables.Reasonable correlation values in the range 0.30.6 should lead to more realistic cost estimates than the overly

10、optimistic values assuming independence or the overly pessimistic values assuming perfect correlation ( Kujawski et al., 2004).Matrix theory implies that a correlation matrix will not have any negative determinants in real life. When a correlation matrix is used in simulation, an important requireme

11、nt is to ensure its feasibility, which restricts the matrix to be positive semi-definite regardless of its type (product-moment or rank) or the way it is estimated (historical or subjective) ( Lurie & Goldberg, 1998). Being positive semi-definite means the eigenvalues of the correlation matrix must

12、be non-negative.That is to say, internal consistency checking between cost elements is necessary for cost estimation. In the literature, it has frequently occurred that the correlation matrix is not positive definite as indicated by Ranasinghe (2000). This is particularly an issue when the number of

13、 dimensions increases because the possibility of having an infeasible correlation matrix will grow rapidly as the dimension increases ( Kurowicka & Cooke, 2001).Tourans approach was to reduce all the correlations slightly (say 0.01) and repeat until the correlation matrix becomes feasible ( Touran,

14、1993). This approach overlooks the possibility of increasing some correlations while reducing others. Ranasinghe (2000) developed a computer program to iteratively calculate and list the bounds of each correlation to make the matrix positive semi-definite. The program then asks the estimator to chan

15、ge the original values and wait until the program re-checks the feasibility and new bounds. This process continues until reaching the feasibility. This approach, however, may be time consuming due to its iterative nature. Yang (2005) developed an automatic procedure to check the feasibility of the c

16、orrelation matrix and adjust it if necessary. It is complicated and difficult to understand due to decomposing the correlation matrix into a diagonal vector of the eigenvalues, and normalization of the diagonal elements to ensure unit diagonals.Here, we advocate that Crystal Ball can be adopted to c

17、onduct the eigenvalue test, on the correlation matrix to uncover this problem. The program warns the user of the inconsistent correlations as Fig. 2.Adjusting the coefficients allows the user to ensure that the correlation matrix is at least not demonstrably impossible. A simple approach to using th

18、e correlation algorithm in the program is to adjust the coefficients permanently after writing down what they were originally. In this way the analyst will find out after the simulation what Crystal Ball had to do to the coefficients to make them possible. This is a minimal test and does not ensure

19、that the correlation coefficients are right in any sense. After examining what the program needed to do, the risk analyst still must take responsibility for the coefficients actually used.4.2. Dilemma for PCA methodologyThe only point value from independent constituent distributions that can be adde

20、d to obtain the corresponding statistical point value from the sum of the constituent distributions is the mean value. Therefore, task-level contingencies derived from individual task distributions cannot be added to obtain the project total contingency.Traditional contingency calculations that add

21、an arbitrary factor to task-level costs and then sum these amounts to a project total, which can produce very conservative project budgets that would be completely outside the calculated distribution of expected results.We review some statistical notations in order to discover the potential problem

22、for typical PCA. For the two random variables x and y, we can have following notations on the basis of probability and statistical theory.4.3. Hierarchy PCA modelDuring the bidding stage, the EPC project must be structured into a limited number of cost items. This does not mean that we will forgive

23、existing detail valuable cost data sets. The reason is that neglecting reliable and valuable cost data sets will influence the efficiency and effect of cost information. Hierarchy probability cost analysis model can be separated into different hierarchies for lower WBS levels.To focus on EPC project

24、s, we assign two hierarchies for hierarchy PCA. The formula (4) is selected for the first hierarchy. We choose WBS level-3 and 4 cost elements for EPC project cost estimation to construct the second hierarchy.4.4. Hierarchy PCA model including MAIMS-PDFsThe MAIMS principle accounts for the fact that

25、 project rarely under-run original allocated budgets. This has important implications for PCA. Once a cost element is allocated a budget x, it be-comes a random variable with minimum value x rather than the lower range Cmin of the original PDF. We refer to these PDFs as the MAIMS-modified PDFs. They

26、 are proper PDFs with a delta-like function at x that accounts for all random values less than or equal to x. We stress the MAIMS modified PDFs with truncated values. Applying the MAIMS principle to a PDF increases its mean value and reduces its standard deviation. The impact increases with increasi

27、ng values of x. The MAIMS principle is likely to play a significant role in PCA. We further investigate this in Section 5 for an actual bidding EPC project.For the hierarchy PCA model including MAIMS-PDFs, we pro-pose all cost elements belonging to the first hierarchy in Section 4.3 will adopt the M

28、AIMS principle, that is to say the budget of baseline will substitute the minimum value of all PDFs of cost elements. The second hierarchy will be same as it is in Section 4.3.4.5. Hierarchy PCA model including hierarchical MAIMS-PDFsIt is easy to find that a baseline budget is necessary for all cos

29、t elements located in first hierarchy, for the hierarchy PCA model including MAIMS-PDFs. As we depict in Section 4.2, to focus on a few vital cost elements and overall influence could be more benefit for cost risk analysis. We present a hierarchy PCA model including hierarchy MAIMS-PDFs to approach

30、this purpose.All PDFs of cost elements in first hierarchy will not have changed in Section 4.3 for hierarchy PCA model including hierarchy MAIMS-PDFs. MAIMS principle is not used for all cost elements5. Practical applicationThe proposed method is applied to the real EPC project to demonstrate its pr

31、actical use. The Beta Pert and Weibull distribution are selected, respectively, for WBS-item cost element based on Section 3. The PDF and relative calibration for WBS level 4 (as Section 3) have done at the beginning of bidding stage with an applicable data set derived from historical cost and exper

32、ts experience.Subjective correlation coefficient method is recommended, and correlation group will be divided based on WBS level 2. The correlation coefficient of pair-wise between cost elements within same group will be assigned as 0.6 as initial coefficients. The correlation coefficient of pair-wi

33、se between cost elements in a different group will be assigned as 0.3 as initial coefficients. The consistency and feasibility of the correlation will be judged and adjusted permanently by Crystal Ball automatically as we point out in Section 4.5.1. Results for four kinds of cost estimation models f

34、or a real bidding EPC project.A simulation experiment is designed to implement the pro-posed method and to evaluate the effects of the hierarchy integrated probability cost analysis model. In the experiment, five kinds of cost analysis models are adopted for assessment. The out-put statistics can th

35、en be used to assess the behavior of the true project cost and the effectiveness of the hierarchy integrated PCA model.5.1.1. Typical PCA modelAll the cost elements and their margin/percentile distributions are shown in Table 1 for typical PCA. The value of each WBS level 3 cost elements is expresse

36、d as kilo euro. This level of granularity is suitable for typical PCA model. Moreover, the WBS level 3 can be changed to reflect the actual situation based on the size of theproject and accuracy of the estimation, if the proposed method is applied to other construction projects.After 5000 simulation

37、 trials, the 6th column in Table 2 lists the descriptive statistics for the total cost of the project. To assess the impact of correlations, we compare two scenarios: including and excluding correlations. 5th column in Table 2 is the result of total project cost including correlations. The first obs

38、ervation is that both distributions are skewed to the right because the mean is larger than the median. The second observation is that the scenario of including correlations has a larger variability (uncertainty) than excluding correlation. This conclusion is unsurprising because the former has a mu

39、ch greater standard deviation that the latter (76687.07 vs. 46474.58, a 35% difference).To select typical down WBS level, the sensitivity analysis is executed and shown in Fig. 4. It is apparent that the unit 13 is more sensitive for the total cost of the project than other units. Unit 13 is selecte

40、d as the 2nd hierarchy for the HPCA cost model.5.1.2. Hierarchy PCA model excluding MAIMS principleAll the cost elements and their margin/percentile distributions for the unit 13 are shown in Table 3.The second hierarchy probability distribution will be generated after 5000 simulation trials, and li

41、sted in Table 4 for details. The new distributions will substitute the corresponding distribution in Table 1. The descriptive statistics for the total cost of the project based on the hierarchy PCA model excluding MAIMS principle, are indicated in the 4th column in Table 2 after 5000 simulation tria

42、ls. The standard deviation of the hierarchy model is not smaller than typical PCA, that is to say, the presented hierarchy model has over-come the impact of increasing cost elements for cost risk analysis.5.1.3. Hierarchy PCA model including MAIMS-PDFsThe descriptive statistics to estimate the total

43、 cost of the project are based on the hierarchy PCA model integrating MAIMS PDF is shown in the 3rd column in Table 2 after 5000 simulation trials.5.1.4. Hierarchy PCA model including hierarchy MAIMS-PDFsThe descriptive statistics for the total cost of the project are based on the hierarchy PCA mode

44、l integrating MAIMS hierarchy is shown in the 2nd column in Table 2 after 5000 simulation trials.5.2. Comparison and validationIn this section, we validate whether the proposed methods can solve the dilemma to appropriate cost elements and maximize the efficiency of cost information for EPC project.

45、 Probability of success (PoS) and confidence internal will be adopted to verify the quality of the estimation.The Monte Carlo simulation result of hierarchy PCA model integrating MAIMS-PDFs is expressed in Fig. 5. The 10% and 90% points of the total cost of the project are based on hierarchy PCA mod

46、el integrating MAIMS-PDFs that establish a 80% confidence interval, and the PoS is generally expressed in percentages of +20.01%/ 14.15%.The Monte Carlo simulation result of hierarchy PCA model integrating MAIMS hierarchy can be expressed as Fig. 6. The 10% and 90% points of the total cost of the pr

47、oject are based on hierarchy PCA model integrating MAIMS hierarchy that established a 80% confidence interval, and the PoS is generally expressed in percent-ages of +20.86%/ 14.61%.The PoS of all models for confidence interval (10%, 90%) is summarized in Table 5. That is to say HIPCA-hierarchy MAIMS

48、-PDFs and HPCA-MAIMS-PDFs can get more realistic cost estimation than typical PCA. The hierarchy PCA model integrated Hierarchy MAIMS-PDFs can achieve more accurate estimation than hierarchy PCA model integrated MAIMS hierarchy.The project baseline cost (PBC) can be concluded as 836 million euro fro

49、m Table 1. The contingency has been summarized in Table 5 based on recommended Practice No. 18R-97 by AACE.All results depict that the HIPCA-hierarchy MAIMS-PDFs and HPCA-MAIMS-PDFs have more realistic and executable estimates. And the proposed methods can solve the dilemma to appropriatecost elemen

50、ts and maximize the efficiency of cost information for EPC project.Finally, cost estimate based on HIPCA-hierarchy MAIMS-PDFs method help us win the bid. The actual reason is such cost estimate is realistic lower, and accompanies with higher PoS. Meanwhile, it provides not only maintain current know

51、ledge of cost overruns, but also estimates cost at completion from inside the project itself rather than by statistical inference from historical information on other projects.6. ConclusionThe practical and theoretically valid hierarchy PCA-hierarchy MAIMS models among WBS-item cost elements have be

52、en developed to solve skillfully the dilemma of typical PCA. The key elements include:1. The use of an appropriate WBS for cost hierarchical structure. Subdividing the project costs into too many bite-size pieces is likely lead to erroneous results and a false sense of confidence. Analysts should be

53、 wary of the pitfalls of performing a probabilistic cost analysis that consists of hundreds of cost elements that are subordinate to WBS-level 3. 2. Macroscopic and microscope risk analysis of project cost elements in order to obtain accurate model input and maximize efficiency of information. Monte

54、 Carlo simulation method is recommended for historical data of WBS level 4 (discipline level) in order to obtain percentile of preliminary PDF. Real estimate of Cmin; Cm; Cmax and reasonable budget will be approached via discipline experts calibration. 3. Incorporation of the money allocated is mone

55、y spent (MAIMS principle) with budget management practices and hierarchy. The assessment of the cost elements, correlation effects, bud-get allocation, and project management consideration items all influence each other and have a significant impact on the total project cost and/or probability of su

56、ccess. For enhanced credibility and realism, HIPCA-hierarchy MAIMS considers these influences simultaneously rather than individually. 4. The proposed approach provides a cost estimation and analysis framework for EPC project. It avoids the impact of high number of cost elements and maximizes effici

57、ency of historical data and experts judgment. And it not only makes demands upon the cost estimator, but also provides benefits to project management, particularly when it comes to recommending a prudent management reserve. Having in hand a probability distribution of total WBS-item cost, rather tha

58、n just a single best estimate, project management can propose, for example, that the basic cost estimate can be budgeted at the 50% confidence level, but that sufficient management reserve can be included to bring the success probability up to 70%. Project managers can develop more viable plans and

59、make better decisions during bidding stage and execution stage, so that projects are delivered for a lower cost and higher probability of success. The magnitude of the cost overrun problem is no excuse for accepting the status quo; the benefits from proposed approaches are likely to be significant.Our experience is that the single greatest challenge to the development a