Double-triangular Distribution is the combination of two triangles, each with an area of 0.5. The mode is also the median.
The mean is:
The variance is:
Note: the variance is the same as for the triangular distribution.
The probability density is:
And the cumulative distribution function:
For Monte Carlo simulation random values from the DT can be generated using random numbers between 0 and 1 (here denoted as “p”) and the following formulas:
The double-triangular is quite unnatural. It is highly unlikely to be a proper representation of uncertainty. Moreover, estimating the median is, in my view, more difficult than the estimating the most likely value (mode).
The three-point estimation technique is used in management and information systems applications for the construction of an approximate probability distribution representing the outcome of future events, based on very limited information. While the distribution used for the approximation might be a normal distribution, this is not always so and, for example a triangular distribution might be used, depending on the application.,
In three-point estimation, three figures are produced initially for every distribution that is required, based on prior experience or best-guesses:
- a = the best-case estimate
- m = the most likely estimate
- b = the worst-case estimate.
These are then combined to yield either a full probability distribution, for later combination with distributions obtained similarly for other variables, or summary descriptors of the distribution, such as the mean, standard deviation or percentage points of the distribution. The accuracy attributed to the results derived can be no better than the accuracy inherent in the 3 initial points, and there are clear dangers in using an assumed form for an underlying distribution that itself has little basis.
Based on the assumption (possibly unwarranted) that a double–triangular distribution governs the data, several estimates are possible. These values are used to calculate an E value for the estimate and a standard deviation (SD) as L-estimators, where:
- E = (a + 4m + b) / 6
- SD = (b − a) / 6
E is a weighted average which takes into account both the most optimistic and most pessimistic estimates provided. SD measures the variability or uncertainty in the estimate. In Project Evaluation and Review Techniques (PERT) the three values are used to fit a Beta distribution for Monte Carlo simulations.
The triangular distribution is also commonly used. It differs from the double-triangular by its simple triangular shape and the mode does not have to coincide with the median. The mean (expectation) is then:
- E = (a + m + b) / 3.
In some applications, the triangular distribution is used directly as an estimated probability distribution, rather than for the derivation of estimated statistics.
The expression “pay it forward” is used to describe the concept of asking the beneficiary of a good deed to repay it to others instead of to the original benefactor. The concept is old, but the phrase may have been coined by Lily Hardy Hammond in her 1916 book In the Garden of Delight.
“Pay it forward” is implemented in contract law of loans in the concept of third party beneficiaries. Specifically, the creditor offers the debtor the option of “paying” the debt forward by lending it to athird person instead of paying it back to the original creditor. Debt and payments can be monetary or by good deeds. A related type of transaction, which starts with a gift instead of a loan, isalternative giving.
he first step in creating a responsibility assignment matrix is to decompose your projectand create a work breakdown structure. Once you have completed this important first step, you will know what the project deliverables will be. If you compose an organizational breakdown structure – breaking the project down into a hierarchy of departments, it will facilitate the process of assigning deliverables to responsible parties. Creating this second chart is an option that is highly recommended.
Once you have the list of deliverables, open an Excel file. Down the left-hand side list each deliverable. If there were intermediate deliverables discovered in the process of creating the work breakdown structure, list those as well.
After listing each deliverable down the side, list each resource across the top of the table.
Now, you will assign deliverables to resources using the following code for roles:
R: Responsible – this is the resource that owns the work. Each deliverable should have at least one person responsible for it.
A: Accountable – this is the person who approves the work. There is only one accountable resource.
C: Consulted – this is the person who delivers information required to complete the work.
I: Informed: This is the person who is informed of the progress of the deliverable.
S: Supportive: This is the person who provides work in addition to the responsible party.
V: Verifies: This is the person who ensures that the work meets standards.
F: Final Authority: This person gives the final stamp on the completed work.
In assigning roles, you will use at least the first four listed above RACI. Assign each deliverable to at least one responsible party. Assign each deliverable exactly one accountable party. Continue until everything has been assigned. Distribute the responsibility assignment matrix monist the staff and make explicit your expectations for each of them.
Earned value management (EVM), or Earned value project/performance management (EVPM) is a project management technique for measuring project performance and progress in an objective manner.
Earned value management is a project management technique for measuring project performance and progress. It has the ability to combine measurements of the project management triangle:
- Schedule, and
In a single integrated system, Earned Value Management is able to provide accurate forecasts of project performance problems, which is an important contribution for project management.
Early EVM research showed that the areas of planning and control are significantly impacted by its use; and similarly, using the methodology improves both scope definition as well as the analysis of overall project performance. More recent research studies have shown that the principles of EVM are positive predictors of project success. Popularity of EVM has grown in recent years beyond government contracting, in which sector its importance continues to rise (e.g., recent new DFARS rules), in part because EVM can also surface in and help substantiate contract disputes.
Essential features of any EVM implementation include
- a project plan that identifies work to be accomplished,
- a valuation of planned work, called Planned Value (PV) or Budgeted Cost of Work Scheduled (BCWS), and
- pre-defined “earning rules” (also called metrics) to quantify the accomplishment of work, called Earned Value (EV) or Budgeted Cost of Work Performed (BCWP).
EVM implementations for large or complex projects include many more features, such as indicators and forecasts of cost performance (over budget or under budget) and schedule performance (behind schedule or ahead of schedule). However, the most basic requirement of an EVM system is that it quantifies progress using PV and EV.
There is a measurement limitation for how precisely EVM can be used, stemming from classic conflict between accuracy and precision, as the mathematics can calculate deceptively far beyond the precision of the measurements of data and the approximation that is the plan estimation. The limitation on estimation is commonly understood (such as the ninety-ninety rule in software) but is not visible in any margin of error. The limitations on measurement are largely a form of digitization error as EVM measurements ultimately can be no finer than by item, which may be the Work Breakdown Structure terminal element size, to the scale of reporting period, typically end summary of a month, and by the means of delivery measure. (The delivery measure may be actual deliveries, may include estimates of partial work done at the end of month subject to estimation limits, and typically does not include QC check or risk offsets.)
A Guide to the Project Management Body of Knowledge (PMBOK Guide) is a book which presents a set of standard terminology and guidelines for project management. The Fifth Edition (2013) is the document resulting from work overseen by the Project Management Institute (PMI). Earlier versions were recognized as standards by the American National Standards Institute (ANSI) which assigns standards in the United States (ANSI/PMI 99-001-2008) and the Institute of Electrical and Electronics Engineers (IEEE 1490-2011).[
he PMBOK Guide is process-based, meaning it describes work as being accomplished by processes. This approach is consistent with other management standards such as ISO 9000 and the CMMI Institute‘s CMMI. Processes overlap and interact throughout a project or its various phases. Processes are described in terms of:
- Inputs (documents, plans, designs, etc.)
- Tools and Techniques (mechanisms applied to inputs)
- Outputs (documents, plans, designs, etc.)
A Guide to the Project Management Body of Knowledge — Fifth Edition provides guidelines for managing individual projects and defines project management related concepts. It also describes the project management life cycle and its related processes, as well as the project life cycle.
The Guide recognizes 47 processes that fall into five basic process groups and ten knowledge areas that are typical of most projects, most of the time.
- The five process groups are:
- Initiating : Those processes performed to define a new project or a new phase of an existing project by obtaining authorization to start the project or phase.
- Planning : Those processes required to establish the scope of the project, refine the objectives, and define the course of action required to attain the objectives that the project was undertaken to achieve.
- Executing : Those processes performed to complete the work defined in the project management plan to satisfy the project specifications
- Monitoring and Controlling : Those processes required to track, review, and regulate the progress and performance of the project; identify any areas in which changes to the plan are required; and initiate the corresponding changes.
- Closing : Those processes performed to finalize all activities across all Process Groups to formally close the project or phase.
- The ten knowledge areas are:
- Project Integration Management : Project Integration Management includes the processes and activities needed to identify, define, combine, unify, and coordinate the various processes and project management activities within the Project Management Process Groups.
- Project Scope Management : Project Scope Management includes the processes required to ensure that the project includes all the work required, and only the work required, to complete the project successfully.
- Project Time Management : Project Time Management includes the processes required to manage the timely completion of the project.
- Project Cost Management : Project Cost Management includes the processes involved in planning, estimating, budgeting, financing, funding, managing, and controlling costs so that the project can be completed within the approved budget.
- Project Quality Management : Project Quality Management includes the processes and activities of the performing organization that determine quality policies, objectives, and responsibilities so that the project will satisfy the needs for which it was undertaken.
- Project Human Resource Management : Project Human Resource Management includes the processes that organize, manage, and lead the project team.
- Project Communications Management : Project Communications Management includes the processes that are required to ensure timely and appropriate planning, collection, creation, distribution, storage, retrieval, management, control, monitoring, and the ultimate disposition of project information.
- Project Risk Management : Project Risk Management includes the processes of conducting risk management planning, identification, analysis, response planning, and controlling risk on a project.
- Project Procurement Management : Project Procurement Management includes the processes necessary to purchase or acquire products, services, or results needed from outside the project team
- Project Stakeholders Management : Project Stakeholder Management includes the processes required to identify all people or organizations impacted by the project, analyzing stakeholder expectations and impact on the project, and developing appropriate management strategies for effectively engaging stakeholders in project decisions and execution.
Each of the ten knowledge areas contains the processes that need to be accomplished within its discipline in order to achieve effective project management. Each of these processes also falls into one of the five process groups, creating a matrix structure such that every process can be related to one knowledge area and one process group.
The PMBOK Guide is meant to offer a general guide to manage most projects most of the time. There are currently two extensions to the PMBOK Guide: the Construction Extension to the PMBOK Guide applies to construction projects, while the Government Extension to the PMBOK Guide applies to government projects. The PMBOK Guide is also used as a support to prepare the Project Management Institute (PMI) certifications, such as the CAPM and PMP.
Strictly speaking matrix management is the practice of managing individuals with more than one reporting line (in a matrix organization structure), but it is also commonly used to describe managing cross functional, cross business group and other forms of working that cross the traditional vertical business units – often silos – of function and geography.
A lot of the early literature on the matrix comes from the field of cross functional project management where matrices are described as strong, medium or weak depending on the level of power of the project manager.
Some organizations fall somewhere between the fully functional and the pure matrix. These organizations are defined in A Guide to the Project Management Body of Knowledge as ’composite’. For example, even a fundamentally functional organization may create a special project team to handle a critical project.
However, today, matrix management is much more common and exists at some level, in most large complex organizations, particularly those that have multiple business units and international operations.
Key advantages that organizations seek when introducing a matrix include:
- To break business information silos – to increase cooperation and communication across the traditional silos and unlock resources and talent that are currently inaccessible to the rest of the organization.
- To deliver work across the business more effectively – to serve global customers, manage supply chains that extend outside the organization, and run integrated business regions, functions and processes.
- To be able to respond more flexibly – to reflect the importance of both the global and the local, the business and the function in the structure, and to respond quickly to changes in markets and priorities.
- To develop broader people capabilities – a matrix helps develop individuals with broader perspectives and skills who can deliver value across the business and manage in a more complex and interconnected environment.
Key disadvantages of matrix organizations include:
- Mid-level management having multiple supervisors can be confusing, in that competing agendas and emphases can pull employees in different directions, which can lower productivity.
- Mid-level management can become frustrated with what appears to be a lack of clarity with priorities.
- Mid-level management can become over-burdened with the diffusion of priorities.
- Supervisory management can find it more difficult to achieve results within their area of expertise with subordinate staff being pulled in different directions.
The advantages of a matrix for project management can include:
- Individuals can be chosen according to the needs of the project.
- The use of a project team that is dynamic and able to view problems in a different way as specialists have been brought together in a new environment.
- Project managers are directly responsible for completing the project within a specific deadline and budget.
The disadvantages for project management can include:
- A conflict of loyalty between line managers and project managers over the allocation of resources.
- Projects can be difficult to monitor if teams have a lot of independence.
- Costs can be increased if more managers (i.e. project managers) are created through the use of project teams.
Representing matrix organizations visually has challenged managers ever since the matrix management structure was invented. Most organizations use dotted lines to represent secondary relationships between people, and charting software such as Visio and OrgPlus supports this approach. Until recently, Enterprise resource planning (ERP) and Human resource management systems (HRMS) software did not support matrix reporting. Late releases of SAP software support matrix reporting, and Oracle eBusiness Suite can also be customized to store matrix information.
The work breakdown structure approach allows us to visually see the work that is needed in order to complete a project.
According to the Project Management Body of Knowledge (PMBOK®), and the Practice Standard for Work Breakdown Structures – Second Edition from the Project Management Institute (PMI), the work breakdown structure can be used to effectively decompose the project scope, to improve estimating, to better control the project execution and to more accurately verify project completion. In addition, using a work breakdown structure approach summarizes project information to improve the opportunity for use of historical information, which, can aid in both speed and accuracy of future projects. The work breakdown structure is a repeatable process that can be used as a template for future projects.
the PMBOK® describes the work breakdown structure as “a deliverable-oriented hierarchical decomposition of the work to be executed by the team”. This is a way of describing the work so that the team knows exactly what work is needed in order to meet the goals of the project. In many cases, a work breakdown structure is the first transition of organizational goals into real work that people can actually perform. It helps to provide clarity of the scope for the project and “breaks down” the scope into whole work units.
A work breakdown structure is deliverable-oriented. So what is a deliverable? In a word, it can best be described as a noun.
A work breakdown structure is a hierarchy. That means that deliverables can be further decomposed into parent and child relationships.
The work breakdown structure technique is described in the Project Scope Management section of the PMBOK and further elaborated in the Practice Standard for Work Breakdown Structures – Second Edition. A scope definition is necessary to decide what all should and should not be covered with in the project. When complete, the work breakdown structure lowest level components called “work packages” can be delegated to teams for further development and estimating and a work breakdown structure dictionary can be developed. In this way, project responsibilities can easily be distributed and committed to. This information can then be placed into a scheduling tool. In addition, the work packages can be used to create a project baseline that assures all the work can be evaluated for completeness.
A work breakdown structure is typically presented in the form of an organization chart-like structure. It can be presented as a list, idea map or outline form as well. The key is to put it in a form that can most easily be used by both the team members and organizational leaders with the tools that your company uses most easily.