Project Management in Three Dimensions
Fundamentally, project management balances three dimensions: the task to be accomplished, the resources dedicated to accomplishing it, and the time allotted. These are generally represented as the three points of a triangle, and their interrelation is immediately clear.
Task
The "task" refers to what is being built, but in a very specific sense: to understand the task, its nature (requirements, features, etc.) must be fully documented, to the point that one can estimate the resources and time needed to accomplish it.
Resources
Resources are items dedicated to the task. It can include people, budget, equipment, material, and more.
Time
Time is simply the schedule for completing the task. It is not an arbitrary decision: the date by which a project can be completed depends of the time it takes to perform the necessary tasks.
All three factors are variables: the more complex the task, the more resources and/or time required to accomplish it; and conversely, the less time or resources, the more the task requirements must be scaled back.
However, mathematics can also be misleading: if you assume that a task that would take one worker ten days could be done by two in five, or ten in an hour. But the task may not be devisable - and even if it is, there will be overhead for coordination and inefficiencies due to conflicting or duplicated efforts. You may find that it takes a ten-man team two hours, or five, or even twelve, to do the same task.
EN: I'm reminded of an analogy: if a person can walk at a speed of four miles an hour, that doesn't mean that five can walk at a speed of twenty miles an hour - the speed of a team is the speed of the slowest person, slower still if you tie their legs together.
Projects in Three Dimensions
Expressing a project in these dimensions provides methods for better management - specifically, in that they are not arbitrary. It is then possible to determine whether a project is feasible (given the task, resources, and time); to demonstrate the consequences of scope creep (increased resources and time in equal proportion); to identify areas where the project will run into snags due to resource availability; etc.
Problem-Solving
There are a myriad of problems that can arise, and a myriad of solutions that can be applied to address them - so the authors cannot provide a panacea, but do suggest a handful of common situations that arise and one method of addressing each (in reality, there are many, and a PM should not limit himself to a default solution).
Bottlenecks (one person must handle many tasks) can be addressed by dividing labor: either get more people to handle the same tasks, or divide the task into components and assign parts of it to different resources.
Crises arise when there is a sudden unexpected change (a key individual leaves the company or management makes arbitrary changes to the scope). Rather than panic, examine the dimensions to determine the net effect and find a solution