Simulation using @Risk - EXPERIMENTAL DESIGN

CHAPTER 3 EXPERIMENTAL DESIGN

3.4 Simulation using @Risk

The spreadsheets in Figure 3.14 and 3.15 are established in order to do simulation with @Risk. The cost data that obtained from previous step are input in mean column that has colored blue in Figure 3.14. And then, input the min and max range of probability distribution data. Cell M6 contains following betageneral formula;

$110 = (80% x $110) + ^{𝜶𝜶𝟏𝟏}

𝜶𝜶𝟏𝟏+𝜶𝜶𝟓𝟓 [(130%x$110)-(80%x$110)], α¹= 52.22

α²= 78.83

Cell K2 (α1) and L2 (α2) are set as adjustable cell with optimization goal target value deterministic cost value (110), the evolver model dialog box is shown in Figure 3.13. Therefore the unknown variable α¹and α²can be determined using Evolver. After the value of α1 and α2 were obtained, those values would be used in the simulation with parameter Betageneral Experiment I (52.55, 78.83, 80%, 130%).

Figure 3.13 Evolver model dialog box to determine α1and α2

Figure 3.14 @Risk BetaGeneral Experiment I (52.55, 78.83, 80%, 130%) spreadsheets (1)

The next step is define distributions, fill the cell cost from column B to I with formula =RiskBetaGeneral(alpha1,alpha2,minimum,maximum) and then add output of simulation in total cost cell B15:I15. Once all of the parameters have been determined, the last step is process the simulation with iterations 10000. The complete simulation output result can be referred in appendixes section.

Figure 3.15 @Risk BetaGeneral Experiment I (52.55, 78.83, 80%, 130%) spreadsheets (2)

In summary, the procedure of time-cost trade-off analysis is as follows.

1. Formulate CPM network each activities in event time,

2. Define all the parameters to be experimented such as interest rates and probabilistic range.

3. Using Evolver to obtain the optimal duration solution from 60 through 76 target duration,

4. Using Evolver to define beta distributions by adjust its probabilistic mean value same as deterministic mean value as well,

5. Simulating the uncertainty model with setting of 10000 iterations and sampling type latin hybercube,

6. Repeat the step (3)and (4) by enlarge distribution range to increase its degree of uncertainty,

7. Classify each simulation output results (5^th percentile values and 95^th percentile values) and,

8. Establish minimum time-cost curve.

CHAPTER 4 RESULTS AND

ANALYSIS

This chapter presents the results of an optimization process and a simulation process. It also presents the analysis of the impact of the time value of money on the time-cost trade-off (TCTO). Furthermore, since this study uses the simulation process, in addition, the probabilistic risk assessment can be analyzed in this chapter.

4.1 TCTO with DFCs Result and Analysis

Table 4.1 shows the total cost of the project for both traditional and DCF analysis of the TCTO problem. As expected, the DCF analysis had a significant impact on the total cost of the project. The optimal project duration (corresponding to the minimum total cost) if the time value of money is ignored (i.e., r = 0%) is 65 weeks, which is identical to that given by Moussourakis and Haksever (2004).

However, if the value starts at 5.25% to 8.50% interest rates, the corresponding optimal project duration would be 71 weeks (Table 4.1).

Figure 4.1 shows the optimal results in a time-cost relationship graph. It is clearly apparent in Figure 4.1 that there is a distinct value for optimal project duration if the time value of money is ignored (65 weeks). This may not be the case for projects with a different network configuration, a different activity utility data, and other values of interest rate.

Figure 4.2 also shows that when higher interest rates (> 8.5%) are applied, the optimum solution duration for interest rates of 12% and 15% is 76 weeks. When the time value of money is ignored, 65 weeks shows as the first optimum duration, 71 weeks shows as the seventh optimum duration, and 76 weeks shows as the twelfth optimum duration. Furthermore, when the time value of money is involved, 71 and 76 weeks show as best optimal durations for some certain value of interest rates. Nevertheless, when this study extends the interest rate, the

optimal duration also changes to 76 weeks for an interest rate of 12%.

Table 4.1. Total cost from optimal solution duration at various interest rates.

Figure 4.1. Total cost solutions for DCFs case with r = 0.00%–8.5%.

Duration

/Interest 0.00% 0.30% 0.50% 1.88% 3.00% 5.25% 6.50% 8.50%

60 $ 973.00 $ 970.51 $ 968.84 $ 957.61 $ 948.71 $ 931.40 $ 922.10 $ 907.66 61 $ 956.00 $ 953.53 $ 951.88 $ 940.77 $ 936.68 $ 914.83 $ 905.63 $ 891.35 62 $ 944.50 $ 941.98 $ 940.30 $ 928.93 $ 919.94 $ 902.45 $ 893.05 $ 878.48 63 $ 933.00 $ 930.42 $ 928.71 $ 917.11 $ 907.93 $ 890.09 $ 878.64 $ 865.66 64 $ 917.00 $ 914.31 $ 912.53 $ 900.39 $ 890.80 $ 872.18 $ 862.19 $ 852.87 65 $ 900.00 $ 897.33 $ 895.58 $ 883.57 $ 874.08 $ 855.65 $ 845.77 $ 830.46 66 $ 901.00 $ 898.27 $ 896.48 $ 884.17 $ 874.45 $ 855.60 $ 845.48 $ 829.82 67 $ 902.00 $ 899.20 $ 897.38 $ 884.77 $ 874.83 $ 855.54 $ 845.20 $ 829.18 68 $ 903.00 $ 900.14 $ 898.28 $ 885.37 $ 875.20 $ 855.48 $ 844.91 $ 828.54 69 $ 904.00 $ 901.07 $ 899.18 $ 885.97 $ 875.57 $ 855.42 $ 844.62 $ 827.91 70 $ 905.00 $ 902.01 $ 900.07 $ 886.57 $ 875.95 $ 855.35 $ 844.33 $ 827.27 71 $ 906.00 $ 902.94 $ 900.97 $ 887.17 $ 876.32 $ 855.29 $ 844.04 $ 826.63 72 $ 907.00 $ 903.91 $ 901.92 $ 887.96 $ 876.99 $ 855.74 $ 844.37 $ 826.78 73 $ 908.00 $ 904.87 $ 902.86 $ 888.75 $ 877.66 $ 856.19 $ 844.70 $ 826.94 74 $ 909.00 $ 905.84 $ 903.80 $ 889.54 $ 878.33 $ 856.63 $ 845.03 $ 827.09 75 $ 910.00 $ 906.80 $ 904.74 $ 890.33 $ 879.00 $ 857.08 $ 845.35 $ 827.24 76 $ 911.00 $ 907.77 $ 905.69 $ 891.12 $ 879.67 $ 862.24 $ 845.68 $ 827.38

Minimum Total Cost

Summary

From the obtained results, it is obvious that ignoring the time value of money in the analysis of the TCTO problem can produce incorrect decisions. The selected duration for activities and, consequently, the optimum project duration depend on the chosen interest rate value and the indirect cost rates. For the sample project on hand, the optimum project duration for DCFs exceeds that of ignoring the time value of money. This case problem example obtained 65 weeks as the optimal duration; however, when interest rates from 5.25% to 8.5% are used, 71 weeks is obtained as the optimal duration instead. This may not be the case for projects with different characteristics. One should therefore be careful in deciding the values of interest rate and of indirect cost.

4.2 TCTO with DCFs in Probabilistic Cost

Section 4.1 provides the optimum project duration with deterministic DCF at different interest rate. This section describes the optimization with probabilistic DCF at

different rates.

4.2.1 Optimization with simulation results analysis

As described in Chapter 3, four experiments were conducted to test the change of optimum total cost at different dispersion of probabilistic values while maintaining the same mean.

For brevity, the following presents the simulation result using @Risk software at interest rates of 0% and 5.25%. Note that 5.25% is the interest rate where the optimum duration changes from 65 weeks to 71 weeks at an interest rate of 5.25%. Table 4.2 shows the statistics of the simulation result when the optimum duration was set to 65 weeks (i.e., the deterministic optimum duration at interest rate of 0.00%) and 71 weeks (i.e., the deterministic optimum duration at interest rate of 5.25%)

For example, at an optimal duration of 65 weeks, r = 0.00%, simulation Experiment I has a 5th percentile value of $892.5, 95th percentile value of

$907.5, and standard deviation value of $4.6 with a skewness value of 0.030, which means that this resulting distribution has a right-skewed distribution—

most values are concentrated on the left of the mean, with extreme values to the right. Furthermore, it also has a kurtosis value of 2.96, which means that a kurtosis value greater than 3 can be categorized as a leptokurtic distribution, sharper than a normal distribution, with values concentrated around the mean and with thicker tails. This means a high probability for extreme values.

Table 4.2. Statistical simulation result of 65 weeks at 0.00% and 71 weeks at 5.25%.

Figures 4.3 and 4.4 show the result of simulation with parameter distribution type betageneral Experiment I (52.55, 78.83, 80%, 130%) in a box plot chart. Figure 4.5 shows that the optimal duration for r = 0.00% is 65 weeks and that for r = 5.25% is 71 weeks, based on their mean values.

However, 67 weeks is its optimal duration at r = 5.25% based on the 5th percentile value, and 71 weeks is its optimal duration based on the 95th percentile value.

Figure 4.5 plots minimum time-cost curves from simulation result of 40

Experiment I and shows the relation between project duration and the simulation output value of total project cost. It obviously shows that ignoring the time value of money in the simulation also gives incorrect optimal duration. Experiment I simulation may not have quite a high degree of uncertainty; therefore, probabilistic optimal duration is quite similar with deterministic optimal duration.

Figure 4.3. Time-cost box plot chart for the simulation results of Experiment I (52.55, 78.83, 80%, 130%) at r = 0%.

Figure 4.4. Time-cost box plot chart for the simulation results of Experiment I (52.55, 78.83, 80%, 130%) at r = 5.25%.

Figure 4.6 shows the results of the simulation from Experiment II, which gives a higher degree of uncertainty of cost by enlarged probability distribution range but still keeps its mean value the same as the deterministic value. Although its optimal duration for the mean remained the same, 71 weeks, its resulting optimal duration for the 5th percentile simulation is 66 weeks, and the 95th percentile simulation results gave 70 weeks as its optimal duration. The detailed results of optimal values in the 5th and the 95th percentile simulation model nos. Experiment I, Experiment II, Experiment III, and Experiment IV can be seen in Table 4.2.

Figure 4.5. Time-cost curve of simulation results for Experiment I at r = 0% and r = 5.25%.

Figure 4.6. Time-cost curve of simulation results for Experiment II with r = 5.25%.

Table 4.3 shows the optimal result of the simulation process for TCTO with DCFs. It shows the optimal duration and total project cost through various interest rates (0.0%–8.5%) and degrees of uncertainty for models of project cost (Experiment I, Experiment II, Experiment III, and Experiment IV). This table presents the simulation results at the 5th and the 95th percentile levels.

Nevertheless, from this table, it could be analyzed that the resulting data obviously show how the different degrees of uncertainty could also produce different optimal solution durations of the TCTO problem. However, at lower interest rate values (0.0%–3.0%), the optimal duration remains the same as the optimal deterministic duration.

Compared with the deterministic result, table 4.3 shows the following:

1. If the cost variation is controlled within 50% as in experiment I, variation of interest rate up to 8.50% does not change the optimal solution.

2. If the cost variation is greater than or equal 80% as in experiment II, III, and IV, the variation of interest rate is equal or greater than 5.25%, it shows a distinct optimal duration with deterministic optimal duration.

The relation of duration–cost–interest rate in each simulation model is graphically illustrated in Figures 4.7, 4.8, 4.9, and 4.10. The graphs imply that adjusting the degree of uncertainty also affects optimal duration in the 5th and the 95th percentile analyses. However, since the value differences were slightly small, the impacts of interest rate parameter adjustment are not illustrated very clearly in the 3-D figures.

Table 4.3. Optimal solution of duration and total project cost at the 5th and the 95th percentile levels.

Figure 4.7. Graph of the cost–time–interest rate relationship Experiment I (52.55, 78.83, 80%, 130%).

0.00% 0.30% 0.50% 1.88% 3.00% 5.25% 6.50% 8.50%

Figure 4.8. Graph of the cost–time–interest rate relationship Experiment II (9.86, 69.04, 90%, 170%).

Figure 4.9. Graph of the cost–time–interest rate relationship Experiment III (5.01, 50.10, 90%, 200%).

Figure 4.10. Graph of the cost–time–interest rate relationship Experiment IV(8.33, 62.49, 80%, 250%).

Summary

One can be obtained from this simulation; it is obvious that involving simulation in order to foresee uncertainty in the analysis of TCTO problem can show other possible decisions. The selected duration for activities and, consequently, the optimal project duration depend on the chosen interest rate value and the premium risk of a decision maker by adjusting its range of probability value.

The different risk profiles of a decision maker may choose different optimal durations from each other. For example, in this case, if the degree of uncertainty is quite high and the decision maker has a risk-averse risk profile, a 70-week duration is chosen as the optimal TCTO duration. However, if the decision maker is optimistic and has a risk-seeker risk profile, a 66-week duration is chosen instead.

Therefore, when the degree of uncertainty in cost is quite high, involving a simulation process in TCTO will obviously give an

unforeseen optimal duration by assuming it to be a deterministic cost.

However, it is necessary to involve DCFs and simulation in TCTO analysis when the interest rate is very low, for instance in some countries such as Japan, Switzerland, and Singapore, and when the degree of uncertainty is not really high as well.

4.2.2 Probabilistic Risk Assessment

Probabilistic risk assessment (PRA) uses probability distributions to characterize variability or uncertainty in risk estimates. In a PRA, one or more variables in the risk equation is defined as a probability distribution rather than a single number. Similarly, the output of a PRA is a range or probability distribution of risks experienced by the receptors. Note that the ability to perform a PRA often is limited by the availability of distributional data that adequately describe one or more of the input parameters.

The primary advantage of PRA is that it can provide a quantitative description of the degree of variability or uncertainty (or both) in risk estimates for total cost project in TCTO problem. The quantitative analysis of uncertainty and variability can provide a more comprehensive characterization of risk than is possible in the point estimate approach.

Another significant advantage of PRA is the additional information and potential flexibility it affords the risk manager. Risk management decisions are often based on an evaluation of high-end risk to an individual for deterministic analyses, this is generally developed by the combination of a mix of central tendency and high-end point values for various exposure parameters.

When using PRA, the risk manager can select a specific upper-bound level from the high-end range of percentiles of risk, in this studies is 95^th percentiles and 5^th percentiles as the low-end range of percentiles of risk.

After finding the optimal solutions, several probabilistic analyses can be performed according to the project planner’s needs. For example, the project planner may want to know the distributions of project cost for a certain solution.

The project planner can use simulation to find this. A simulation is performed by using the suggested ‘‘best options duration’’ presented in Table 3.4.

From the previous simulation, 65 weeks was obtained as optimal duration for interest rate 0.00% and 71 weeks were obtained as optimal duration for interest rate 5.25%; therefore this part takes 65 and 71 weeks to analyze probability risk assessment. Figure 4.11, 4.12, and 4.13 illustrate the probability density function (PDF) and cumulative distribution function (CDF) of the project cost at optimal duration, respectively.

We can estimate the project cost for this example by using the mean value.

In addition, the project planner can look into the CDF of the project duration to find out the probability that the project can be finished within a certain desired cost and degree of uncertainty.

Figure 4.11 Probability density function of total project cost Experiment I at duration 65 weeks (r = 0.00%) and duration 71 weeks(r =

5.25%)

Figure 4.12 Cumulative distribution function of total project cost Experiment I at duration 65 weeks at r = 0.00%

Figure 4.13 Cumulative distribution function of total project cost Experiment I at duration 71 weeks at r = 5.25%

The result according to 4.13 for 71 weeks Experiment I could be interpretable as follows:

1. The mean figure for total cost at this optimal duration will be

$855.29.

2. The minimum figure for total cost at this optimal duration will be $ 838.41, but this figure is the bottom line and will only be achieved if all positive circumstances would occur. Therefore the implementation of Value at Risk (VaR) is necessary. The result for VaR 5% is $ 848.47. That means with a probability of 5 %, the figure for total cost at this optimal duration will not exceed $ 848.47.

Or in other words: with a probability of 95 % the figure for total cost will exceed $ 848.47.

3. The maximum figure for total cost at this optimal duration will be

$870.53. However, this figure is the upper limit and will only be achieved if all negative circumstances would occur. Therefore the implementation of Value at Risk (VaR) is also necessary under this point of view. The result for VaR 95% is $862.02. That means with a probability of 95 %, the figure for total cost at this optimal duration will not exceed $862.02. Or in other words: only with a probability of 5 % the figure for total cost at this optimal duration will exceed

$862.02.

Figure 4.11 and 4.12 show the PDF and CDF of the project cost with four types of degree uncertainty, respectively.For example, if the desired duration is 71 days, interest rate 5.25%, degree uncertainty betageneral Experiment II at cost $860, the probability would be around 79.6%.

Summary

Using a probabilistic approach to arrive at optimum project duration not only makes better use of the available historical data (if available) or risk profile, but it also provides additional information to facilitate the planning process associated to the reduction of a project schedule. Being able to doing PDA is one of the most important

advantages from the simulation process. A decision-maker can obtain the optimal duration at various percentile levels of the risk results. Nevertheless, by more complete provided information for uncertainty, cost also helps the decision-maker of the project to solving TCTO with DCFs under uncertainty cost condition; therefore, incorrect optimal duration can be avoided.

Managing risks in construction projects has been recognized as a very important management process in order to achieve the project objectives in terms of time, cost, quality, safety and environmental sustainability. The uncertainty of the cost project depends on the risk factor projects. Furthermore, the probability range of uncertainty cost is identified based on cost planner’s risk analysis. Table 4.4 shows the risk factors of the construction project that influence in identification of the range probability.

Table 4.4 Risk factor of construction project (Karim, 2012)

CHAPTER 5 CONCLUSION

5.1 Conclusion

Most of the time-cost trade-off (TCTO) analysis in construction management has disregarded a time value of money. From the results obtained, it was obvious that ignoring a time value of money in the analysis of TCTO problem could produce incorrect decisions. The proposed approach can help the practitioners in considering net present value in time-cost decisions leading to identification of the best option. It can be concluded that Discounted Cash Flows (DCFSs) should be considered during the analysis of TCTO problem, especially for projects span over time periods more than one year. TCTO analysis with DCFs produces realistic results and consequently, sound decisions.The proposed approach can help the practitioners in considering net present value in time-cost decisions leading to identification of the best option. It can be concluded that DCF should be considered in the analysis of TCTO problem, especially for projects span over time periods more than one year rather than projects span less than one year that is not effective to be considered. TCTO analysis with DCF produces realistic results and consequently sound decisions.

This research utilizes simulation techniques to imitate the probabilistic nature of project networks throughout the search of optimal solutions. This approach provides more realistic solutions for construction time-cost trade-off problem under uncertainty. An example project from the literature is used to demonstrate the usefulness of the model and to illustrate its capabilities. The model has the following features: (1) optimum solution is guaranteed; (2) precise discrete activity time-cost relationship is used; (3) time value of money is taken into consideration; and (4) uncertainty in the project also are involved. The results show that inclusion of discounted cash flow results in distinct optimal project duration. Nevertheless, through the Monte Carlo simulation, which is in

order to involve uncertainty, this proposed model also lead to distinct optimal duration at the certain percentile level. It also demonstrates that risk analysis techniques such as probabilistic risk assessments that can provide more unforeseen probability risk of assessing project time and cost risks in TCTO problem. This approach provides construction engineers with a new way of analyzing construction time/cost decisions in a more realistic manner with considering a time value of money and degree of uncertainty. The proposed model for this study can help the practitioners in considering time value of money and uncertainty cost, in order to make the best time-cost decision and to identify risks involved.

5.2 Recommendation for future research

A computational optimization model has been developed, which links the CPM with least-cost optimization, DCFs techniques and uncertainty cost in order to optimize the traditional Time-Cost Trade-Off problem. Although the main theme of proposed model is to account for DCFs, effect of inflation can be incorporated in further studies. Moreover, the stochastic interest rate model can also be incorporated in further studies. The model can be extended to handle

在文檔中 Discounted Cash Flows Time-Cost Trade-Off Problem Optimization with Uncertainty Cost (頁 43-0)