## Discrete Random Variables

So far, our examples of random variables have been discrete random variables. H or T could only take on discrete values on any specific toss 1 or 0. On any given toss, we have no way of knowing what value H or T will take, but we can estimate or calculate what the probable outcomes are, and we can say for certain, because the random variables are discrete, that they will not take on in-between values. For sure, H cannot take on a value of 0.75 on any specific toss only values of 1 (true) or 0...

## Continuous Random Variables

As the number of values of X increases in a given range of values, the spacing between them becomes smaller, so small in the limit that one cannot distinguish between one unique value and another. So also do the value's individual probabilities become arbitrarily small in order not to violate the rule about all probabilities adding up to 1. Such a random variable is called a continuous random variable because there is literally no space between one value and another one value flows continuously...

## Summary of Important Points

Table 7-2 provides the highlights of this chapter. Table 7-2 Summary of Important Points Table 7-2 Summary of Important Points The major program milestones that mark business value drive the project at the highest level. The program milestones come from the business case. Program milestones do not have a probabilistic character. The program logic is found in the detail project schedules. Lower level schedules are in network form, preferably the precedence diagramming method, and tie together...

## Some Characteristics of the Critical Path

We have so far described the critical path as the longest path through the network. This is true and it is one of the clearest and most defining characteristics of the critical path. A second idea is that there is no float or slack along the critical path. Having no float or slack means that if there is any change in durations along the critical path, then the overall schedule will be longer or shorter. In effect, such a characteristic means there is no schedule reserve that can isolate...

## Integrating the Project Balance Sheet and Business Value Models

Now let us integrate the concepts discussed in this chapter to complete the framework on which we hang quantitative analysis to be discussed in the remainder of this book. The business models drive the left side of the project balance sheet. The model results, working through the value flow-down process, frame opportunity (new products, new markets and customers, operational and organizational needs) and quantify goals. Goals, deployed through strategy, lead to identified projects. It remains...

## The Central Limit Theorem Applied to Networks

Take notice that the critical path through the network always connects the beginning node or milestone and the ending node or milestone. The ending milestone can be thought of as the output milestone, and all the tasks in between are input to the final output milestone. Furthermore, if the project manager has used three-point estimates for the task durations, then the duration of any single task is a random variable best represented by the expected value of the task. 2 The total duration of the...

## Monte Carlo Simulation of the Network Performance

The arithmetic of finding expected value, standard deviation, and variance, at least to approximate values suitable and appropriate to project management, is not hard to do when working with the most common distributions we have described so far in this book. Anyone with reasonable proficiency in arithmetic can do it, and with a calculator or spreadsheet the math is really trivial. However, the manual methodology applied to a network of many tasks, or hundreds of tasks, or thousands, or even...

## Work Breakdown Structure Baseline

For quantitative estimating purposes, we need a target to shoot at. It is better that the target be a static target, not moving during the time of estimation. The scope captured on the WBS at the time estimating begins is the project baseline scope. Baselines are fixed until they are changed. Changing the baseline is a whole subject unto itself. Suffice to say that once resources are assigned to a baseline, and then that baseline is changed, the task of the project manager and project...

## Multiple Independent Variables

Having more than one independent variable complicates calculations immediately. The dependent variable, say cost, now depends on two different and independent variables, say schedule and worker productivity. Much of the conceptual ground is the same. Indeed, the r2 becomes R2, but the idea remains the same the measure in a figure of merit of how the dependent data are driven by the independent data. For the more complex projects there may be a need to do multiple variate regression. The only...

## Economic Value Add and Net Present Value Equivalence

Fortunately for project managers, NPV and EVA are exactly equivalent. NPV is computationally EVA-NPV equivalence is a very big convenience indeed. Let's see how this equivalence works ii shows the calculations. PV of EAT present value of earnings after tax. PV of CCE present value of the cost of capital employed. PV of EVA present value of economic value add. PV of NCF present value of net cash flow. First, we must reorient ourselves to cash flow rather than earnings. Tom Pike's ditty Cash is...

## The Trial Balance and the Work Breakdown Structure

Everything on the WBS must roll up to some account on the chart of accounts. Assuming that the project administrator has a mapping of the WBS accounts to the chart of accounts, the WBS can be reconciled to the trial balance using pivot queries in a spreadsheet. t1lRike, Tom, Retool, Rethink, Results, Simon & Schuster, New York, 1999. 2lHiggins, Robert C., Analysis for Financial Management, Irwin McGraw-Hill, Boston, MA, 1998, chap. 1, pp. 16-21. 3lHiggins, Robert C., Analysis for Financial...

## Calculating the Regression Curve

Up to this point, we have discussed single independent variable regression, albeit with the cart in front of the horse we discussed the linear equation before we discussed the data observations. In point of fact, the opposite is the case in real projects. The project team has or makes the data observations before there is a curve. The task then becomes to find a curve that fits the data. E31 There is plenty of computer tool support for regression analysis. Most spreadsheets incorporate the...

## Quantitative Techniques in Time Management

Time management is amply described in most texts on project management. In this chapter we will focus only on the quantitative aspects of time management and make a couple of key assumptions to begin with The major program milestones that mark real development of business value are identified and used to drive the project at the highest level. The program logic is found in the detail project schedules. Lower level schedules are in network form, preferably the precedence diagramming method (PDM)...

## The Cash Flow Statement

Now we come to the cash and to the flow of cash. As described by Robert Higgins in his book, Analysis for Financial Management, t2 the cash flow statement is commonly thought of as the place to put down the sources of cash and uses of cash in the project. Viewed this way as a double entry system, the sources ought to balance the uses. Flow refers to a change over time. As such, a cash flow statement is not a statement of cash on hand, but rather the change in cash over the reporting period....

## The Expense Statement

The P& L expense statement is one of the three most important financial statements that the controller will provide to the project manager. The other two financial statements that are useful to project managers, as were discussed in Chapter 5, are the cash flow statement and the balance sheet. The project manager has little to say about the expense categories on the P& L the controller usually defines the expense categories when the chart of accounts is put together. On the other hand,...

## The Mathematics of Project Contracts

Mathematical formulas enter the picture when the project manager seeks to dollar-quantify the risk transferred or retained by a contracting activity. Each contract type, whether FP or CP, has a set of mathematical parameters. Table 9-1 provides a summary of the major contract types and the principal financial parameters for each. In the following paragraphs, we will present examples of how these parameters are applied to various contracts. Let's begin with FFP. Table 9-1 Financial Parameters in...

## The p Space

To this point, we have been using a number of conventions adopted for probability analysis All quantitative probabilities lie in the range between the numbers 0 (absolute certainty that an outcome will not occur) and 1 (absolute certainty that an outcome will occur). The lower case p is the notation for probability it stands for a number between 0 and 1 inclusively. Typically, p is expressed as a decimal. If p is the probability that project outcome A will happen, then 1-p is the probability...

## Cost Categories on the Profit and Loss Statement

Cost categories on the P& L can be interlaced in a hierarchy according to the view most advantageous to managers. Direct expenses could be divided into fixed and variable, or you could turn it around so that fixed expenses could be divided into direct and indirect. At the bottom line, the sum total is indifferent to the management's selection of the cost hierarchy. flThe exception to the project manager creating the WBS occurs when the project is on a contract for a customer who chooses to...

## K

Figure 7-1 Schedule Building Blocks. i> e nurk.Hri on schedule, it S not i> e nurk.Hri on schedule, it S not Figure 7-2 Milestone Simplifications. Figure 7-2 Milestone Simplifications. There are certain characteristics that are applied to each schedule task, represented by a rectangle in our network building blocks. These task characteristics are Every task has a specific beginning and a specific ending, thereby allowing for a specific duration (ending...

## The Normal Distribution

The Normal distribution is a well-known shape, sometimes referred to as the bell curve for its obvious similarity to a bell. In some texts, it will be referred to as the Gaussian distribution after the 19th century mathematician Carl Friedrich Gauss. The Normal distribution is very important generally in the study of probability and statistics and useful to the project manager for its rather accurate portrayal of many natural events and for its relationship to something called the Central Limit...

## Net Present Value and Net Future Value

You may recognize that discounting is the inverse of the familiar idea of compounding. Compoui and forecasts a future value based on an interest rate or capital factor rate. Discounting begins w which a discount factor is applied to obtain a present value of the amount. In the same project, t discount rate are the same rate. Therefore, it is relatively easy to work from the present to the fu present. Most of us who have had a savings or investment account are familiar with the compounding for...

## The Treacy Wiersema Model

Michael Treacy and Fred Wiersema described a model of business value in their study, Customer Intimacy and Other Value Disciplines, 6 published in the Harvard Business Review, and expanded further in their book, The Discipline of Market Leaders. 7 Closely aligned with the balanced scorecard, the Treacy-Wiersema model has three focus areas. The first is customer intimacy, in which the concept of relationship management as a business value is foremost. Customer intimacy is characterized by a...

## Forecasting with the Time Centric System

Just like in the traditional earned value system, forecasts can be made using the same formula as we developed for the forecast in that system Forecast Actual performance + Remaining performance Index For example, in the first period the actual starts are 3, the remaining performance for the project is 17, and the index is 0.6. The forecast is therefore First period index 3 5 0.6 Actual starts 3 Forecast 3 + 17 0.6 3 + 28.3 31.3 starts where 31.3 equivalent starts. How should the project...

## The Work Breakdown Structure

The WBS has been a fixture of project management for many years. 31 The WBS is simply the organizing structure of the entire scope of the project. There is one organizing rule that governs overall if an item is on the WBS, then that item is within the scope of the project if an item is not on the WBS, then that item is out of scope of the project. The WBS identifies the deliverables and activities that are within the scope of the project. The WBS is most typically shown as an ordered...

## Decision Trees with Dependent Conditions

With Bayes' Theorem in hand, we can proceed to project decisions that are interdependent. Let' is an item on the WBS that may or may not be included by the sponsor's decision in the final pro buy decision for satisfying the acquisition to be made by the project team. However, let's change decision by the sponsor affects the subsequent performance of either make or buy Let SD be the random variable that represents a sponsor's decision that may or may not be affects subsequent make or buy...

## Three Point Estimate Approximations

Quite useful results for project statistics are obtainable by developing three-point estimates that can be used in equations to calculate expected value, variance, and standard deviation. The three points commonly used are Most pessimistic value that yet has some small probability of happening. Most optimistic value that also has some small probability of happening. Most likely value for any single instance of the project. The most likely value is the mode of the distribution. It is not...

## Time and Materials Contract Math

A commonly employed contract vehicle for obtaining the services of temporary staff or to engage in highly speculative R& D is the time and materials (T& M) contract. The usual form of this contract is that the time charges are at a standard and fixed rate for a labor category, but there may be many different labor categories, each with a different labor rate, that are chargeable to the contract. Nonlabor items for all manner of material, travel, subsistence, and other things are charged...

## Project Rules for Claiming Earned Value Credit

There are three rule systems generally applied to earned value systems (1) all or nothing, such as was applied in the bicycle project example (2) some number of discrete steps like 0, 50, or 100 and (3) continuous estimates of credit from 0 to 100 complete with any number in between being acceptable. Practice has shown that the continuous method dilutes the value of the earned value method since the tendency to be 90 complete for most of the timeline is very prevalent. Of course it is not...

## Earned Value Equations for Variances and Indexes

From the bicycle example, we have developed some experience with the two most important earned value equations that address project history the cost variance and the schedule or value variance. Table 6-4 provides all the equations associated with the earned value measurements that are important to project managers. You can see that they are quite simple mathematically. In this table you will find not only the equations that address history but also the equations necessary to understand the...