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Probabilistic Model for Cost Contingency Ali Touran, M.ASCE1 Abstract: This paper proposes a probabilistic model for the calculation of project cost contingency by considering the expected number of changes and the average cost of change. The model assumes a Poisson arrival pattern for change orders and independent random variables for various change orders. The probability of cost overrun for a given contingency level is calculated. Typical input values to the model are estimated by reviewing several U.S. Army Corps of Engineers project logs, and numerical values of contingency are calculated and presented. The effect of various parameters on the contingency is discussed in detail. DOI: 10.1061/共ASCE兲0733-9364共2003兲129:3共280兲 CE Database subject headings: Probabilistic methods; Models; Cost control; Project management. Introduction In many construction projects, the owner plans for unexpected events that may affect project cost by adding a contingency to the estimated cost. Depending on the owner’s organization and level of sophistication, this contingency is calculated in various ways. One of the most simple and common methods is to consider a percent of estimated cost, such as 10%, based on previous experience with similar projects. Such an approach will not quantify the degree of confidence that the contingency will provide against cost overruns. In this paper we introduce a probabilistic model that incorporates uncertainties in project cost and calculates the contingency based on the level of confidence specified by the owner. The model’s input parameters are few, and the application is straightforward. Model Let us assume that events causing cost adjustments 共change orders兲 occur randomly in time. Assuming that these events happen according to a Poisson process, we have e ⫺ x P 关 X⫽x 兴 ⫽ ; x⫽0,1,2,... (1) x! where X is a random variable denoting the number of incidents during the project, and is the mean of the distribution and is calculated from ⫽␣T (2) where ␣ is the mean rate of occurrence and T is the estimated project duration. We further assume that each incident will affect the project cost. Specifically, the cost of change order i is assumed to be C I , which is a random variable. The total cost of changes C ch will be 1 Associate Professor, Dept. of Civil and Environmental Engineering, Northeastern Univ., 400 Snell Engineering Center, Boston, MA 02115. Note. Discussion open until November 1, 2003. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on January 25, 2002; approved on May 28, 2002. This paper is part of the Journal of Construction Engineering and Management, Vol. 129, No. 3, June 1, 2003. ©ASCE, ISSN 0733-9364/2003/3280–284/$18.00. x C ch ⫽ 兺 Ci (3) i⫽1 The probability mass function 共PMF兲 of total cost of changes can now be computed. Derivation of Distribution for PMF of C ch We consider two cases: in the first case we assume that change orders are distributed according to a normal distribution; in the second case this assumption is relaxed. Special Case If one assumes that cost of change orders are independent and identically distributed normal variables, then C i ⬃N 共 C , C2 兲 (4) In the above equation, C and C are the mean and standard deviation of change order cost, respectively. Total cost of changes is normally distributed because it is a linear sum of x normal variables, where x is itself a random variable, as it represents the number of incidents causing cost adjustments in a construction project n for x⫽n⇒C ch 兩 x⫽n ⫽ 兺 Ci (5) i⫽1 From the above equation, one can calculate conditional distributions for C ch and the expected value of C ch as follows: C ch 兩 x⫽n ⬃N 共 n C ,n C2 兲 冋兺 册 (6) x E 关 C ch 兴 ⫽E i⫽1 C i ⫽E 关 X 兴 C Now, referring to Poisson distribution for X, we have e ⫺ n P 关 X⫽n 兴 ⫽ P 关 C ch ⫽N 共 n C ,n C2 兲兴 ⫽ n! (7) (8) General Case At this point we can relax the requirement that all costs are identical; the equations presented above will change but the approach remains the same C i ⬃N 共 C i , C i 兲 (9) 280 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003 Table 1. Projects’ Cost and Schedule Data Description Original duration 共cda兲 Original budget 共$兲 Number of changes Cost of changes 共$兲 Regional finance center Concrete repairs Replace gate system Construct seepage control Meadow brook restoration Flood damage reduction 730 150 360 730 120 730 4,664,369 139,283 1,127,000 16,360,130 413,954 6,774,400 37 4 12 10 7 25 1,382,721 71,132 172,320 712,761 364,456 1,159,187 Project code DACA51-98-C-0033 DACW33-00-C-0014 DACW33-95-C-0037 DACW33-97-C-0018 DACW33-97-C-0019 DACW33-97-C-0021 a Calendar days. 冉兺 n C ch 兩 x⫽n ⬃N i⫽1 n Ci, 兺 i⫽1 C2 i 冊 (10) We now relax the requirement that all change costs are normally distributed. Eq. 共10兲 still holds because even if C i ’s are not normally distributed, according to the central limit theorem the sum of these will be normal, given that i is sufficiently large. Several references discuss the magnitude of i. References in applied statistics such as Devore 共2000兲 recommend values larger than 30, but suggest that if individual variables are close to bellshaped, even a small value for i will suffice. Others, mostly engineering sources 关such as Moder et al. 共1983兲兴, suggest that with i being as small as four, still the assumption of normality is approximately acceptable. For this specific application, the number of changes for a project with a duration of between 12 and 24 months would be somewhere between two cases. The main problem with the ‘‘general case’’ is that it would be difficult to estimate parameters of n delays and n cost distributions with any degree of certainty. It should be noted that the cost increase calculated above is based on a specific , the total number of change orders. If project changes cause an increase in project duration, the expected number of changes will increase accordingly and will impact the cost of change. In this model, this increase in cost is disregarded partially because the contingency is estimated at the beginning of the project as a function of original budget. Calculation of Contingency As Percentage of Original Cost It is convenient to express contingency as a percentage of the original estimate or budget. We express contingency as the ratio of total contingency  over original budget C. Consider the following: C⫽original project cost estimate 共budget excluding contingency兲; T⫽original project duration estimate 共duration excluding contingency兲; and f C ⫽coefficient of variation for change in cost due to change orders. C⫽ f C C Calculation of Contingency Now that we have the distribution of cost overruns, we can specify a desired contingency value. Here, for simplicity, we again assume that changes are identical and independent. If the owner desires a confidence level of p against cost overruns 共that is, the probability of not having a cost overrun being p兲, he or she would need a contingency of  such that P 关 C ch ⭐ 兴 ⭓p (11) ⬁ P 关 C ch ⭐ 兴 ⫽ 兺 x⫽0 ⬁ ⫽ P 关 C ch ⭐ 兩 X⫽x 兴 P 关 X⫽x 兴 兺⌽ x⫽0 冋冑 册 ⫺x C e ⫺ x ⭓p x! x 2 O i⫽ (13) 冉 2 2 C f C C Ci ⇒O i ⬃N , C C C2 O ch 兩 x⫽n ⬃N 冉 n C n f C C , C C2 2 2 冊 冊 (14) (15) In the above equations, O i denotes the cost of a change order as a ratio of total original project cost, and O ch is the total cost adjustment as a result of changes in the project. Using the following equations, we can calculate percent contingencies for the budget. Note that changes are assumed to be identical and independent (12) P 共 O ch ⭐ 兲 ⭓p C (16) Table 2. Projects’ Input Values and Statistical Parameters Project code DACA51-98-C-0033 DACW33-00-C-0014 DACW33-95-C-0037 DACW33-97-C-0018 DACW33-97-C-0019 DACW33-97-C-0021 Mean of change order 共$兲 Standard deviation of change order 共$兲 Coefficient of variation of change order Mean of change as ratio of original estimate 共%兲 Cost of changes as ratio of original estimate 共%兲 Rate of change per month Level of significance for exponential distribution 共%兲 53,400 62,800 21,200 72,800 52,100 43,000 131,000 40,800 28,100 70,200 63,200 50,900 2.44 0.65 1.32 0.97 1.21 1.19 1.14 45.1 1.9 0.4 12.5 0.6 28 51 15 4 88 17 1.5 0.8 1 0.4 1.8 1 9 ⬍1 ⬍1 1 30 ⬍1 JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003 / 281 Fig. 1. Distribution of time between changes for projects studied ⬁ P 共 O ch ⭐ 兲 ⫽ 兺 x⫽0 ⬁ ⫽ more, the expected value of O ch can be calculated. It can be shown that if a random variable can be modeled as the sum of X random variables, where X is a random variable itself, we have 共Benjamin and Cornell 1970兲 P 关 O ch ⭐ 兩 X⫽x 兴 P 关 X⫽x 兴 兺⌽ x⫽0 冋冑 册 ⫺ x C C x f C2 C2 X e ⫺ x x! ⭓p Y⫽ (17) C2 Eq. 共17兲 gives the probability of the total cost of changes remaining below any assumed contingency percentage . Further- 兺 Zi (18) i⫽1 E 关 Y 兴 ⫽E 关 X 兴 •E 关 Z 兴 (19) Var关 Y 兴 ⫽Var关 Z 兴 •E 关 X 兴 ⫹ 兵 E 关 Z 兴 其 2 •Var关 X 兴 (20) From the equations above one can calculate parameters of O ch 282 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003 E 关 O ch 兴 ⫽E 关 X 兴 •E 关 O i 兴 ⫽E 关 X 兴 • c c ⫽ C C Var关 O ch 兴 ⫽Var关 O i 兴 •E 关 X 兴 ⫹ 兵 E 关 O i 兴 其 •Var关 X 兴 ⫽ 2 f C2 C2 C2 (21) ⫹ C2 C2 (22) Examination of Assumptions and Calculation of Input Values To develop numerical values for a contingency, one needs to establish typical ranges for the input variables; and model assumptions also need to be evaluated. The main statistical assumptions in the model are the following: change orders arrive according to a Poisson process, and the cost of change orders are independently and identically distributed normal variables. If the number of changes is sufficiently large, say, larger than 10 共Miller 1983兲, then the assumption of normality for individual change orders can be relaxed. Regardless of the distribution of changes, the total cost of change will have an approximately normal distribution. In order to evaluate the Poisson assumption made in the development of the model, we examined data for several projects obtained form the U.S. Army Corps of Engineers, New England Division. The Corps provided us with project change logs of 34 projects that had either cost or schedule changes. Of the 34 projects investigated, only 6 were selected for statistical analysis. The reasons for eliminating the others were that 7 of the projects had no change orders, 13 had fewer than four change orders 共mostly projects of very limited scope and duration兲, and 8 had some important information missing, such as ‘‘notice to proceed’’ or the value of some of the change orders 共Davul 2001兲. The 6 projects that were examined in detail consisted of building, dredging, and drainage projects 共Table 1兲. We also used this data to calculate typical values for inputs to the model such as average change size, coefficient of variation for changes, and number of changes. Fig. 2. Probability of cost overrun given contingency of 15% and various values of f c and c (⫽12) a sufficient number of changes for the assumption of normality for the total cost of changes. The main inputs to the model are mean change size ( c ); coefficient of variation ( f c ); and total number of change orders 共兲. Reasonable ranges for these variables were selected based on observed data. For example, for the Army Corps of Engineers 共ACE兲 projects, one may consider a range of 0.5 to 2.5 for the coefficient of variation, 0.5 to 2% of the original project estimate for average change, and a rate of 0.5 to 1.5 changes per month 共Table 2兲. With the model calibrated, one can perform various sensitivity analyses and investigate the effect of each variable on potential cost overruns and required contingencies. As an example, Fig. 共2兲 shows that the probability of a cost overrun is much more sensitive to variability of average change size compared to f c . In fact, one may be able to make a case by stating that f c may be taken as a fixed value; a sufficient contingency can be estimated based on two random factors: c and 共estimated number of change orders during project execution兲. Statistical Analysis We assumed that changes happen according to a Poisson process, which means that the time between changes is distributed according to an exponential distribution. The test of chi-square goodness of fit was performed on the 6 projects, and the results are provided in Table 2. As can be seen, the level of significance for 3 of the projects is less than 1%, which shows that while an exponential distribution may not be a good fit for presentation of data in all cases, it is appropriate in a good proportion of cases 共in this experiment, in 3 out of 6 projects兲. Fig. 1 gives the histogram for the time between changes for all 6 projects. As can be seen, although the test results were not encouraging for all cases, the exponential distribution in general seems to provide a reasonable visual fit for the data. Input Data for Model A convenient way of using the model developed in this study is to provide a tabular or graphical solution to Eq. 共17兲. In order to achieve this, a MATLAB 共Hanselman and Littlefield 1999兲 routine was created whose results are reported later in this paper. Inputs to the model were selected based on parameters presented in Table 2. For this case, we have limited the application to projects with durations between 12 and 24 months, which ensured Application and Analysis of Results Using the ACE data, the model is calculated and the probability of a cost overrun for various assumed contingency levels is calculated. For this experiment, a fixed f c ⫽1.0 is selected because, as shown earlier, the probability of a cost overrun is less sensitive to variations of f c . The result of the analysis for a 12-month project is presented in Table 3. Assuming that a confidence level of 75% is desired with respect to contingency allocation, Table 3 highlights the combinations of and ␣ 共rate of change per month兲 that result in a cost overrun 共insufficiency of contingency兲. Fig. 3 shows the result of the analysis graphically. The expected value of the total change can be calculated using Eq. 共21兲 as (12␣) c /C. As an example, if the average change order is 1% of the original estimate, then the expected value of cost of changes, given one change per month, would be 12⫻1⫻1% ⫽12%. This means that a contingency of 12% should be adequate about 50% of the time. While one can use average values and perform a deterministic analysis for the contingency, the model provides confidence levels when the amount of contingency deviates from the expected values. Increasing the contingency will reduce the probability of a cost overrun. JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003 / 283 Table 3. Probability of Cost Overrun with 15% Contingencya ␣ 0.5 0.6 0.7 0.8 0.9 1.0 a c/C ⫽0.5% c/C ⫽1% c/C ⫽1.5% c/C ⫽2% 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.03 0.07 0.11 0.18 0.26 0.13 0.22 0.32 0.43 0.54 0.64 0.31 0.43 0.56 0.66 0.75 0.82 Project duration is assumed to be 12 months. Finally, to see the effect of contingency variation on the probability of a cost overrun, the value of contingency was ranged and the probability of a cost overrun was calculated as a function of a contingency percentage 共Fig. 4兲. This is equivalent to the graphical solution of Eq. 共17兲. Using Fig. 4, the owner can choose a level of contingency commensurate with a risk acceptable to him on her. In this figure, it is assumed that the coefficient of variation f C ⫽1 and C ⫽1%. Fig. 4. Probability of cost overrun for various values of contingency for various number of change orders ( C ⫽0.01 and f C ⫽1) Future Work A probabilistic model for the calculation of contingency is presented and its potential demonstrated by applying typical input values; a similar model can be developed for schedule contingency. Interaction of schedule delays and cost increases is another area that deserves further research. Also, an extensive survey of various project types can be conducted to calculate typical input values for specific types of projects. As an example, a transit agency is usually engaged in specific types of construction projects. By reviewing the historical data of a specific transit agency, one can calculate rates of changes, size and distribution of changes, and times between changes for similar projects and prepare risk profiles or cumulative probability curves for various values of contingency. The outcome can be used at the budgeting phase of a new project to ensure that consideration is given to potential cost overruns after the project starts. Acknowledgments The writer would like to thank Paul Cooper of the Army Corps of Engineers, New England Division, for his assistance in providing project cost data. Atilla Davul, a graduate student in the Civil Engineering Department, Northeastern University, helped in data analysis. References Fig. 3. Probability of cost overrun given contingency budget of 15% of original estimate and T⫽12 months for various rates of change orders per month Benjamin, J. R., and Cornell, C. A. 共1970兲. Probability, statistics, and decision for civil engineers, McGraw-Hill, New York. Davul, A. 共2001兲. ‘‘Analysis of the construction cost data.’’ MS thesis submitted to Northeastern Univ., Boston. Devore, J. L. 共2000兲. Probability and statistics for engineering and the sciences, 5th Ed., Duxbury, Pacific Grove, Calif. Hanselman, D., and Littlefield, B. 共1999兲. MATLAB user’s guide, Mathworks Inc., Natick, Mass. Miller, R. W. 共1983兲. Schedule, cost, and project control with PERT, McGraw-Hill, New York. Moder, J. J., Phillips, C. R., and Davis, E. W. 共1983兲. Project management with CPM, PERT and precedence diagramming, Van Nostrand Reinhold, New York. 284 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003