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The Project Management Triangle (called also the Triple Constraint, Iron Triangle and "Project Triangle") is a model of the constraints of project management. While its origins are unclear, it has been used since at least the 1950s. It contends that:
For example, a project can be completed faster by increasing budget or cutting scope. Similarly, increasing scope may require equivalent increases in budget and schedule. Cutting budget without adjusting schedule or scope will lead to lower quality.
In practice, however, trading between constraints is not always possible. For example, throwing money (and people) at a fully staffed project can slow it down. Moreover, in poorly run projects it is often impossible to improve budget, schedule or scope without adversely affecting quality.
The Project Management Triangle is used to analyze projects. It is often misused to define success as delivering the required scope, at a reasonable quality, within the established budget and schedule. The Project Management Triangle is clearly insufficient as a model of project success because it omits crucial dimensions of success including impact on stakeholders, learning and user satisfaction.
The time constraint refers to the amount of time available to complete a project. The cost constraint refers to the budgeted amount available for the project. The scope constraint refers to what must be done to produce the project's end result. These three constraints are often competing constraints: increased scope typically means increased time and increased cost, a tight time constraint could mean increased costs and reduced scope, and a tight budget could mean increased time and reduced scope.
Another approach to project management is to consider the three constraints as finance, time and human resources. If you need to finish a job in a shorter time, you can throw more people at the problem, which in turn will raise the cost of the project, unless by doing this task quicker we will reduce costs elsewhere in the project by an equal amount.
As a project management graphic aid, a triangle can show time, resources, and technical objective as the sides of a triangle, instead of the corners. John Storck, a former instructor of the American Management Association's "Basic Project Management" course, used a pair of triangles called triangle outer and triangle inner to represent the concept that the intent of a project is to complete on or before the allowed time, on or under budget, and to meet or exceed the required scope. The distance between the inner and outer triangles illustrated the hedge or contingency for each of the three elements. Bias could be shown by the distance. His example of a project with a strong time bias was the Alaska pipeline which essentially had to be done on time no matter the cost. After years of development, oil flowed out the end of the pipe within four minutes of schedule. In this illustration, the time side of triangle inner was effectively on top of the triangle outer line. This was true of the technical objective line also. The cost line of triangle inner, however, was outside since the project ran significantly over budget.
James P. Lewis  suggests that project scope represents the area of the triangle, and can be chosen as a variable to achieve project success. He calls this relationship PCTS (Performance, Cost, Time, Scope), and suggests that a project can pick any three.
The real value of the project triangle is to show the complexity that is present in any project. The plane area of the triangle represents the near infinite variations of priorities that could exist between the three competing values. By acknowledging the limitless variety, possible within the triangle, using this graphic aid can facilitate better project decisions and planning and ensure alignment among team members and the project owners.
The STR model is a mathematical model which views the "triangle model" as a graphic abstraction of the relationship:
Scope refers to complexity (which can also mean quality). Resources includes humans (workers), financial, and physical. Note that these values are not considered unbounded. For instance, if one baker can make a loaf of bread in an hour in an oven, that doesn't mean ten bakers could make ten loaves in one hour in the same oven (Due to the oven capacity).
For analytical purposes, the time required to produce a deliverable is estimated using several techniques. One method is to identify tasks needed to produce the deliverables documented in a work breakdown structure or WBS. The work effort for each task is estimated and those estimates are rolled up into the final deliverable estimate.
The tasks are also prioritized, dependencies between tasks are identified, and this information is documented in a project schedule. The dependencies between the tasks can affect the length of the overall project (dependency constrained), as can the availability of resources (resource constrained). Time is different from all other resources and cost categories.
Using actual cost of previous, similar projects as the basis for estimating the cost of current project.
According to the Project Management Body of Knowledge (PMBOK) the Project Time Management processes include:
Due to the complex nature of the 'Time' process group the project management credential PMI Scheduling Professional (PMI-SP) was created.
To develop an approximation of a project cost depends on several variables including: resources, work packages such as labor rates and mitigating or controlling influencing factors that create cost variances. Tools used in cost are, risk management, cost contingency, cost escalation, and indirect costs . But beyond this basic accounting approach to fixed and variable costs, the economic cost that must be considered includes worker skill and productivity which is calculated using various project cost estimate tools. This is important when companies hire temporary or contract employees or outsource work.
Project management software can be used to calculate the cost variances for a project.
Requirements specified to achieve the end result. The overall definition of what the project is supposed to accomplish, and a specific description of what the end result should be or accomplish. A major component of scope is the quality of the final product. The amount of time put into individual tasks determines the overall quality of the project. Some tasks may require a given amount of time to complete adequately, but given more time could be completed exceptionally. Over the course of a large project, quality can have a significant impact on time and cost (or vice versa).
Together, these three constraints have given rise to the phrase "On Time, On Spec, On Budget." In this case, the term "scope" is substituted with "spec(ification)."
Traditionally the Project Constraint Model recognised three key constraints; "Cost", "Time" and "Scope". These constraints construct a triangle with geometric proportions illustrating the strong interdependent relationship between these factors. If there is a requirement to shift any one of these factors then at least one of the other factors must also be manipulated.
With mainstream acceptance of the Triangle Model, "Cost" and "Time" appear to be represented consistently. "Scope" however is often used interchangeably given the context of the triangle's illustration or the perception of the respective project. Scope / Goal / Product / Deliverable / Quality are all relatively similar and generic variation examples of this, while the above suggestion of 'People Resources' offers a more specialised interpretation.
This widespread use of variations implies a level of ambiguity carried by the nuance of the third constraint term and of course a level of value in the flexibility of the Triangle Model. This ambiguity allows blurred focus between a project's output and project's process, with the example terms above having potentially different impetus in the two contexts. Both "Cost" and "Time" / "Delivery" represent the top level project's inputs.
The 'Project Diamond' model  engenders this blurred focus through the inclusion of "Scope" and "Quality" separately as the 'third' constraint. While there is merit in the addition of "Quality" as a key constraining factor, acknowledging the increasing maturity of project management, this model still lacks clarity between output and process. The Diamond Model does not capture the analogy of the strong interrelation between points of the triangles however.
PMBOK 4.0 offered an evolved model based on the triple constraint with 6 factors to be monitored and managed. This is illustrated as a 6 pointed Star that maintains the strength of the triangle analogy (two overlaid triangles), while at the same time represents the separation and relationship between project inputs/outputs factors on one triangle and the project processes factors on the other. The star variables are:
When considering the ambiguity of the third constraint and the suggestions of the "Project Diamond"; it is possible to consider instead the Goal or Product of the project as the third constraint, being made up of the sub factors "Scope" and "Quality". In terms of a project's output both "Scope" and "Quality" can be adjusted resulting in an overall manipulation of the Goal/Product. This interpretation includes the four key factors in the original triangle inputs/outputs form. This can even be incorporated into the PMBOK Star illustrating that "Quality" in particular may be monitored separately in terms of project outputs and process. Further to this suggestion, the use of term "Goal" may best represent change initiative outputs, while Product may best represent more tangible outputs.