Project Management
Risk Management and Optimisation Examples
ISBN: 9781522527039
Author: Vojo Bubevski (Independent Researcher)
Abstract
The Results
The complete results of the improved Efficient Frontier portfolios are as follows. Table 5 shows the Mean Return (µ) and the Investment Fractions (F) of total funds invested in each project.
Table 5: The Investment Fractions (F) in Improved Portfolios
µ |
0.15 |
0.155 |
0.16 |
0.165 |
0.17 |
0.175 |
0.18 |
0.185 |
0.19 |
F P1 |
0 |
0 |
0 |
0 |
0 |
0 |
0.0201 |
0.0623 |
0.266 |
F P2 |
0.0199 |
0.0349 |
0.0498 |
0.0695 |
0.0974 |
0.1249 |
0.1521 |
0.1775 |
0.041 |
F P3 |
0.1806 |
0.1761 |
0.1708 |
0.1600 |
0.1374 |
0.1141 |
0.0714 |
0 |
0 |
F P4 |
0.1289 |
0.1250 |
0.1202 |
0.1123 |
0.0983 |
0.0833 |
0.0505 |
0 |
0 |
F P5 |
0 |
0 |
0.0031 |
0.0023 |
0 |
0 |
0 |
0 |
0 |
F P6 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
F P7 |
0.1949 |
0.2183 |
0.2419 |
0.2685 |
0.2997 |
0.3317 |
0.3601 |
0.3848 |
0.225 |
F P8 |
0.1900 |
0.2132 |
0.2357 |
0.2607 |
0.2896 |
0.3181 |
0.3458 |
0.3754 |
0.469 |
F P9 |
0.1048 |
0.0651 |
0.0249 |
0 |
0 |
0 |
0 |
0 |
0 |
F P10 |
0.1809 |
0.1674 |
0.1536 |
0.1267 |
0.0776 |
0.0279 |
0 |
0 |
0 |
Total |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
µ |
0.15 |
0.155 |
0.16 |
0.165 |
0.17 |
0.175 |
0.18 |
0.185 |
0.19 |
V |
0.0070 |
0.0075 |
0.0081 |
0.0087 |
0.0095 |
0.0104 |
0.0114 |
0.0128 |
0.0161 |
σ |
0.0834 |
0.0865 |
0.0898 |
0.0934 |
0.0975 |
0.1022 |
0.1072 |
0.1130 |
0.1270 |
SR |
1.1568 |
1.1736 |
1.1855 |
1.1931 |
1.1941 |
1.1890 |
1.1793 |
1.1637 |
1.0748 |
The improved Efficient Frontier of the Minimal Mean-Variance portfolios is presented herein. The Efficient Frontier curve shows that an increase in the expected return of the portfolio causes an increase in the portfolio Standard Deviation. Once again, the Efficient Frontier gets flattered as expected. This shows that each additional unit of Standard Deviation (i.e., risk) allowed, increases the portfolio Mean Return by less and less.
ISBN: 9781522527039
The presented method is applied to Project Management by using PERT-CPM critical paths to manage project risk. DMAIC framework applies proven stochastic techniques as follows: 1. Define: Stochastic optimisation determines the critical paths; 2. Measure: Every critical path is simulated and associated risks are calculated. Six Sigma process metrics are calculated against specified targets; 3. Analyse: Simulation results are analysed and sensitivity analysis is used to identify and quantify the main contributors to the variability of the project duration time; 4. Improve: The critical paths are ranked and prioritised for management’s attention based on their associated risk factors; 5. Control: The project was not implemented, so there is no data for analysis. However, assuming that the project was implemented, a generic Project Control phase is applied.
Keywords: Operations Research; Project management; PERT-CPM; Six Sigma; DMAIC; Stochastic Optimisation; Monte Carlo simulation.
The Results
Stochastic optimisation is used to resolve the critical paths of the project. We minimise Project Duration Time (15), with the constraint that the Activity Time Elements (VATE) must be equal or greater than the Activity Estimated Duration (AED) (13). The project start time T(1) is initialised to zero. The project time values T(2) – T(8) are adjustable and calculated by the model to achieve the optimum, subject to the constraint that they are positive numbers (2). For the optimisation, we need to specify trial values for the adjustable project times, so we specify zero to all project time values T(2) – T(8).
Table 2: Stochastic Optimisation Results for the Project’s 10 Activities
Activity Id & Description |
Activity Time Element (Months) |
Estimated Duration (Months) |
Activity Slack Time (Months) |
1 Design Plant |
12 |
12 |
0 |
2 Select Site |
9 |
9 |
0 |
3 Prepare Site |
13 |
13 |
0 |
4 Select Vendor |
4 |
4 |
0 |
5 Produce Generator |
18 |
18 |
0 |
6 Install Generator |
4 |
4 |
0 |
7 Prepare Manuals |
12 |
5 |
7 |
8 Select Personnel |
16 |
3 |
13 |
9 Train Operators |
10 |
9 |
1 |
10 Obtain License |
6 |
6 |
0 |
Table 3: Stochastic Optimisation Results for the Project’s Paths
Path (Critical Indicator) |
Path Activities |
Path Total Duration (Months) |
Path Slack Time (Months) |
1 (Yes) |
1-2-3-6-10 |
44 |
0 |
2 (Yes) |
1-4-5-6-10 |
44 |
0 |
3 (No) |
1-4-7-9-10 |
36 |
8 |
4 (No) |
1-8-9-10 |
30 |
14 |
Path 3 has a slack of 8 months and includes three critical activities, i.e., activities 1, 4, and 10; and two non-critical activities, i.e., activities 7 and 9. This means that the non-critical activities on this path can suffer a total of 8 months of delays before affecting the completion of the project.
Similarly, Path 4 has a slack of 14 months and includes two critical activities, i.e., activities 1 and 10; and two non-critical activities, i.e., activities 8 and 9. This means that the non-critical activities on this path can suffer a total of 14 months of delays before affecting the completion of the project.
Chapter 1: Improving Project Critical Path
In Book 3: Stochastic and Six Sigma Improvements to Project Risk Management
ISBN: 978-620-2-02905-6
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter elaborates on how to reduce the risks in a project to construct a power plant by applying PERT-CPM. There are ten activities in the project with empirically estimated duration and cost. Dependencies amongst activities are defined, so a network diagram is created, determining the critical path and project duration. Two simulations are run, one by one, to determine project duration, cost, and associated risks. Simulation-1 applies the empirically estimated durations and costs as inputs. Then Simulation-2 is run by applying the Simulation-1 result values as inputs. The results are analysed and evaluated. Also, sensitivity analysis is performed to identify and quantify the main contributors to variability and risk. It is established that the Simulation-2 solution is superior, importantly achieving a reduction in the project risk, which is the objective.
Keywords: Project risk management; PERT-CPM; Critical path; DMAIC; Financial risk management; Monte Carlo simulation; Six Sigma; Slack; Stochastic model.
The Results
Simulation 1 and Simulation 2 results including the project duration mean and the associated risk factors are presented in Table 6. The two simulations are compared considering Mean Duration (µ) in months (m), Standard Deviation (σ), and the Six Sigma process capability metrics: Process Capability (Cp), Process Capability Index (Cpk), and Sigma Level (σ-L). The comparison shows that the Simulation 2 results for the project duration are slightly increased, but the associated risk factors are improved against the Simulation-1 risk factors. This acknowledges that the risk is reduced in Simulation-2, which improved and benefited the project.
Table 6: Project Duration and Risk Factors
|
µ (m) |
σ |
Cp |
Cpk |
σ-L |
Simulation-1 |
48.37 |
5.216% |
0.2907 |
-0.286 |
0.2611 |
Simulation-2 |
49.7 |
4.738% |
0.3397 |
0.0989 |
0.8220 |
Table 7: Project Cost and Risk Factors
|
µ ($) |
σ |
VAR |
Cp |
Cpk |
σ-L |
Simulation-1 |
6,180,000 |
4.3634% |
90.885% |
0.3572 |
-0.1372 |
0.4629 |
Simulation-2 |
6,313,330 |
4.1256% |
90.858% |
0.3955 |
0.2248 |
1.0525 |
ISBN: 978-620-2-02905-6
The Results
The original project’s planned total time and cost, and simulations’ total time & cost including risk factors are presented in Table 5 to provide for comparison and evaluation of results. Comparing the results of Initial-Plan’s Simulation-1, the project’s total time and cost significantly deviate from Task-22-Completed-Plan’s original, including actual values for 22 tasks; the differences are -203.77 (hrs) and -$4,653 for time and cost respectively. However, Simulation-2 of Initial-Plan results deviate much less from Task-22-Completed-Plan’s original values; the differences are only -17.45 (hrs) and -$1,174 for time and cost respectively. Even though Standard Deviations and VAR are insignificantly increased, Simulation-2 results are superior and more realistic.
Comparing the results of Simulation-2 of Initial-Plan and Simulation-3 of Task-22-Completed-Plan, the project total time and cost of both simulations deviate insignificantly from Task-30-Completed-Plan’s original values, including actual values for 30 tasks. That is: i) Simulation-2 differences are 17.45 (hrs) for time and $1,174 for cost; and ii) Simulation-3 differences are 37.6 (hrs) for time, and $1,257 for cost, i.e., slightly higher than Simulation-2. However, Simulation-3 slightly reduced the total project time and cost, significantly reduced the Standard Deviation, and increased, i.e., improved, the process capability metrics, which are the major risk factors. From a Project Risk Management perspective, this is the ultimate objective, even though VAR is increased insignificantly. Therefore, Simulation-3 results are superior and more realistic.
Author: Vojo Bubevski
Abstract
This chapter presents a stochastic analysis in order to evaluate projects in terms of both, risk and return. If the projects provide the same return, the aim is how to evaluate the projects’ risk. and identify which project is riskier. Simulation is used to calculate the projects’ return and variance. The objective is to resolve the problem specified above. The evaluation determines the project with a smaller risk, which is the objective.
Keywords: Project Risk Management, Risk Analysis, Stochastic Model, Monte Carlo Simulation, Six Sigma DMAIC
The Results
In the Improve stage, the risk factors of the simulation results are compared in order to acknowledge which risk factors are better, which is the objective of this analysis. Project Return and Risk Factors are compared in Table 3.
Table 3: Project Return Risk Factors
|
µ (m) |
σ |
Cp |
Cpk |
σ-L |
Project 1 Return |
10% |
1.000% |
0.6667 |
0.6667 |
2.0000 |
Project 2 Return |
10% |
3.266% |
0.2041 |
0.2041 |
0.5791 |
Risk factors of the two projects’ returns are compared considering the Mean (µ), Standard Deviation (σ), Process Capability (Cp), Process Capability Index (Cpk), and Sigma Level (σ-L). The comparison shows that the project returns are the same, but the risk factors of Project 1 Return are much better than Project 2 Return risk factors. This acknowledges that Project 2 is significantly riskier than Project 1. The Risk Difference of Return Equivalents risk factors is compared in Table 4.
Table 4: Risk Difference of Return Equivalents Risk Factors
|
µ ($) |
σ |
Cp |
Cpk |
σ-L |
Risk Neutral Difference |
0% |
3.421% |
0.3897 |
0.3897 |
1.1115 |
Risk Averse Difference |
1.662% |
6.416% |
0.2078 |
0.1215 |
0.8921 |
Risk factors of the Risk Difference of Return Equivalents are compared considering the same factors as above. The comparison shows that the Risk Neutral Difference is 0%, and the Risk Averse Difference is 1.662%. The risk factors of Risk Neutral Difference are much better compared to the Risk Averse Difference risk factors. It is clear that Risk Averse Difference is significantly riskier than Risk Neutral Difference.
ISBN: 978-620-2-02905-6
Author: Vojo Bubevski (Independent Researcher)
The Results
The simulation results of project Cash Flows for the years 2012 – 2021 are presented in Table 1 and Table 2 below.
Table 1: Cash Flows in USD ($) Results for 2012 – 2016
Year |
2012 |
2013 |
2014 |
2015 |
2016 |
Project Cost |
-11,550 |
-223,550 |
-64,100 |
0 |
0 |
Initial Revenue Year |
0 |
0 |
0 |
1 |
1 |
Revenue |
0 |
0 |
0 |
140,000 |
154,000 |
Revenue Adjustment |
0 |
0 |
0 |
-59,452 |
0 |
Fixed cost |
0 |
0 |
0 |
-49,000 |
-49,000 |
Variable cost |
0 |
0 |
0 |
-70,000 |
-77,000 |
Cash Flow |
-11,550 |
-223,550 |
-64,100 |
199,548 |
280,000 |
Year |
2017 |
2018 |
2019 |
2020 |
2021 |
Project Cost |
0 |
0 |
0 |
0 |
0 |
Initial Revenue Year |
1 |
1 |
1 |
1 |
1 |
Revenue |
169,400 |
186,340 |
204,974 |
225,471 |
248,019 |
Revenue Adjustment |
0 |
0 |
0 |
0 |
0 |
Fixed cost |
-49,000 |
-49,000 |
-49,000 |
-49000 |
-49000 |
Variable cost |
-84,700 |
-93,170 |
-102,487 |
-112,736 |
-124,009 |
Cash Flow |
303,100 |
328,510 |
356,461 |
387,207 |
421,028 |
The Cash Flow starts at -$11,550 in 2012 and drops to a minimum of -$223,550 in 2013. Then it continuously increases from the minimum in 2013 to $421,028 in 2021.
The simulation Output Values results are shown in Table 3.
Table 3: Simulation Output Values Results
Output Values |
LSL-Lower |
USL-Upper |
TV-Target |
VAR at 1% |
|
NPV ($) |
865,609.63 |
594,188.25 |
1,029,926.30 |
792,251 |
411,834.79 |
IRR (%) |
0.58009 |
0.3975 |
0.689 |
0.53 |
0.39929 |
Total Variable Cost ($) |
664,101.97 |
498,076.50 |
863,332.60 |
664,102 |
311,827.29 |
Summary Stats |
|||||
Mean NPV ($) |
871,813.52 |
||||
Probability NPV negative |
0 |
ISBN: 978-620-2-02905-6
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates optimisation of project cost and duration by minimising the Net Extra Cost. The project is a residential site construction managed in Microsoft™ Project. Simulation and stochastic optimisation models consider uncertainty in tasks’ durations, to calculate the project’s duration, cost, bonuses, penalties, and Net Extra Cost. Firstly, simulation calculates the tasks durations, the project total duration, cost, bonuses, penalties, and the Net Extra Cost, including the associated risks. Then, stochastic optimisation recalculates the project’s total duration and cost by minimising the Net Extra Cost. Simulation and optimisation results are analysed to evaluate the project’s total cost and duration, and the Net Extra Cost, including the related risk factors. It is concluded that the total cost and duration of the project are significantly decreased by the optimisation, with negligible change to the associated risks, which is the objective.
Keywords: Project Risk Management, Project Cost and Duration, Six Sigma DMAIC, Stochastic Optimisation, Monte Carlo Simulation
The Results
In the Improve stage, the simulation results are compared with the optimisation results (Table 3) in order to acknowledge if an improvement has been achieved.
Table 3: Simulation vs. Optimisation Results
Variable |
Calculated |
Risk-Mean |
Simulation Project Total Cost ($) |
63,325.42 |
60,462.80 |
Optimisation Project Total Cost ($) |
53,949.25 |
57,912.54 |
Simulation Project Total Duration (Weeks) |
28.36667 |
26.6667375 |
Optimisation Project Total Duration (Weeks) |
25.00 |
26.6666647 |
Simulation Project Total Duration (Days) |
141.83333 |
133.333688 |
Optimisation Project Total Duration (Days) |
125 |
133.333323 |
Simulation Net Extra Cost ($) |
4,876.17 |
6,052.82 |
Optimisation Net Extra Cost ($) |
-4,500.00 |
-1,200.0 |
Project Total Cost VAR @ 1% |
VAR ($) |
VAR (%) |
Simulation |
54,431.74 |
90.0251768 |
Optimisation |
51,687.00 |
89.2495841 |
ISBN: 978-620-2-02905-6
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter applies a stochastic model to a petroleum water flood project in order to estimate recoverable oil, and to select one of four observed production schedules which generate the maximal revenue stream. The stochastic optimisation model combines volumetric estimates, prices, costs, and production scheduling. The objective is to maximise the Cash Flow, Net Present Value (NPV), and Internal Rate of Return (IRR) for the project over an 18-year horizon. The model considers specified information about initial costs, operating costs, reservoir description, production schedules, prices, working interest, and taxes.
Keywords: Project Risk Management, Petroleum Water Flood, Production Scenarios, Cash Flow, Net Present Value (NPV), Internal Rate of Return (IRR), Six Sigma DMAIC, Stochastic Optimisation, Monte Carlo Simulation.
The Results
The DCF (Discounted Cash Flow) summary trend for years 1 – 18 shows: Cash Flow for Years 1 – 3 is $0.00 MM. From Year 3, it quickly increases, reaching a maximum of around $17.50 MM in Year 7. It then slowly reduces to zero in Year 13 and remains zero till Year 18. The Outputs results are shown in Table 1.
Outputs |
Calculated Value |
VAR at 1% |
NPV ($) |
70,269,701.36 |
33,241,051.16 |
IRR (%) |
0.35351 |
VAR (%) |
Total Reserves (STB) |
2,967,435 |
47.305 |
Project NPV and IRR results are presented in Table 2. The optimisation results for NPV and IRR are compared with the four simulations results considering Mean NPV (µ), Standard Deviation (σ) and Value at Risk (VAR).
Table 2: NPV and IRR Optimisation vs. Simulations’ Results
|
NPV µ $ |
NPV σ $ |
NPV VAR $ |
IRR µ % |
IRR σ % |
Optimisation |
70,245,803 |
19,603,286 |
33,421,051 |
34.855 |
5.292 |
Production Schedule 1 Sim |
70,279,369 |
19,734,323 |
32,791,218 |
34.857 |
5.329 |
Production Schedule 2 Sim |
63,114,682 |
18,056,019 |
28,839,777 |
30.669 |
4.917 |
Production Schedule 3 Sim |
57,826,651 |
16,801,838 |
25,905,879 |
28.078 |
4.694 |
Production Schedule 4 Sim |
52,669,256 |
15,579,575 |
23,128,530 |
25.543 |
4.466 |
ISBN: 978-620-2-02905-6
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter elaborates on how to determine the optimal timing for a corporation’s project portfolio. Stochastic optimisation is applied to schedule starting years of 10 projects in the portfolio. Stochastic optimisations are used to determine the starting years of the projects that maximise the Net Present Value (NPV) of cash flows from all the projects in the portfolio. Two optimisations are run to identify two local optimums. The two optimal solutions are compared in order to select and implement the preferred timing of the projects. In this chapter, the optimal solution with better NPV and IRR values, as well as a smaller risk, is selected. However, the selection of the preferred solution is subject to corporations’ management decisions. It should be noted that in a real-world situation, more than two local optimums are required, i.e., as many as practical, to make the decision on the optimal timing of the implementation of projects.
Keywords: Project Risk Management; Project Portfolio, Projects timing, Six Sigma, DMAIC, Monte Carlo simulation
The Results
In the Improve stage, the Optimisation 1 vs. Optimisation 2 results are presented in Table 5 for comparison.
Table 5: Optimisation 1vs. Optimisation 2 Comparison
Outputs |
Calculated Value |
VAR at 1% |
VAR (%) |
Optimisation 1 NPV ($MM) |
2,054.108 |
1,551.16 |
75.515 |
Optimisation 2 NPV ($MM) |
1,962.6 |
1,561.75 |
79.576 |
Optimisation 1 IRR (%) |
35.351 |
25.85 |
73.124 |
Optimisation 2 IRR (%) |
32.443 |
27.146 |
83.673 |
ISBN: 978-620-2-02905-6
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter presents a stochastic cost analysis for a company’s portfolio of ten projects over 12 months. Each project is planned to start in a given month, and from that month on, it has expected costs, some of which are known (of which, some are zero) and some of which are uncertain. There are possible random delays (or, for a few projects, possible earlier starting months), which shift the cost schedule to the right (or left). Also, each project has a variable chance of failing (i.e., 5%, 10%, 15%, or 20%) in any month after its actual starting month. If it fails in a given month, that month’s costs plus any remaining months’ costs are not incurred. A simulation model is applied to calculate the variables that are required for the analysis. Four simulations are run to complete the analysis.
Keywords: Project Risk Management; Cost Analysis, Project Portfolio, Six Sigma, DMAIC, Monte Carlo simulation
The four sets of simulation results are presented in Table 5 for comparison.
Simulation No |
Failure Probability |
Total Cost ($M) |
T. Cost Delta ($M) |
Standard Dev. ($M) |
Std. Dev. Delta ($M) |
Simulation 1 |
5% |
13,831.79 |
N/A |
1,266.76 |
N/A |
Simulation 2 |
10% |
12,440.24 |
1,392 |
1,583.58 |
317 |
Simulation 3 |
15% |
11,204.04 |
1,236 |
1,718.41 |
134 |
Simulation 4 |
20% |
10,115.58 |
1,088 |
1,749.96 |
32 |
ISBN: 978-620-2-67095-1
Author: Vojo Bubevski (Independent Researcher)
Abstract
The Results
The simulation results are presented below. The attained values for NPV, Net NPV, Net NPV Mean, and Probability of Net NPV Negative are given in Table 1.
Table 1: Simulation Output Results
Data |
Value |
NPV |
$128,884,418 |
Net NPV |
$36,600,000 |
Net NPV Mean |
$39,660,000 |
Probability of Net NPV Negative |
14.9875% |
ISBN: 978-620-2-91949-4
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates a stochastic optimisation of a project portfolio considering the projects’ capital budgeting with financial statements. There are eight proposed projects to be selected for the optimal portfolio, bearing in mind the projects’ Cash Flows, Net Present Value (NPV), Total Expenses, and Present Cost, for a period of 10 years in the future. The financial statements of the projects are simulated including several uncertain variables. Each project has a chance of falling from a technical standpoint. If it fails, it is not part of the total portfolio, that is, its NPV and costs are not included. The objective is to select the projects for the portfolio that maximise the mean portfolio NPV but to ensure with 95% confidence that the total present value of costs stays below the budget.
Keywords: Risk Assessment and Management, Project Portfolio, Optimisation, Financial Risk, Sensitivity Analysis; Monte Carlo simulation; Stochastic model.
The Results
The calculated Portfolio Cash Flow (CF) per Year of all projects is presented in Table 4.
Table 4: Portfolio Cash Flow per Year ($)
Year |
2021 |
2022 |
2023 |
2024 |
2025 |
CF |
-543,333 |
-526,358 |
798,941 |
275,965 |
670,355 |
Year |
2026 |
2027 |
2028 |
2029 |
2030 |
CF |
650,508 |
954,850 |
1,237,922 |
965,937 |
1,030,113 |
Year |
2021 |
2022 |
2023 |
2024 |
2025 |
TE |
543,333 |
700,733 |
433,455 |
321,264 |
264,162 |
Year |
2026 |
2027 |
2028 |
2029 |
2030 |
TE |
247,153 |
250,239 |
206,528 |
189,706 |
191,388 |
ISBN: 978-620-2-91949-4
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter presents a stochastic model for estimating the costs of a new project. When submitting a budget proposal for a project, two key questions need to be answered: 1. What is the probability that the project will actually be delivered within this budget? 2. How much contingency (i.e. extra budget) should be included in order for this new revised budget level to be achieved with a certain degree of confidence? A simulation model is created to estimate the costs and calculate the required variables to answer these questions. It is assumed that each item’s actual cost will be within a min-max range. To involve uncertainties, probability distributions are applied to describe the possible costs of each item in practice. The distributions are assumed to be skewed in this case, and the parameters chosen to reflect those costs are assumed to more likely overshoot than undershoot the base case. The cost elements of the project are the model’s inputs, which are projections of historical data available.
Keywords: Risk Assessment and Management, Project Budget Proposal, Project Costs Estimation, Financial Risk, Sensitivity Analysis; Monte Carlo simulation; Stochastic model.
The Results
The Total Project Cost was calculated with Monte Carlo Simulation. The Total Project Cost Mean is $18,962.50 M with a Standard Deviation of $488.15 M, i.e., 2.574%, which is a very low risk. The Six Sigma target parameters are TV = $18,963.0 M; LSL = $18,500.0 M; and USL = $19,500.0 M. The Six Sigma process capability metrics are Cp = 0.3414; Cpk = 0.3158; and σ-L = 0.4369. There is a 5% probability that the cost will be below $18,181 M; a 90% probability that it will be in the range of $18,181 M – $19,7.