The Critical Value Algorithm for Selecting a Distribution Function

CRITICAL VALUE AND PERT

We present a methodology for eliciting a distribution for commuting time of a driver. The individual example presents an approach to estimating a distribution function mostly when no historic data is available, and only the expert's opinion is available.

The variable in question here is to select the best distribution for the driving time of a certain commuter who provides information about his/her usual driving times during workings days at the usual hour at which he/she usually commits to work.

USING A PERT DISTRIBUTION

If (1) there is no data, (2) no other distribution is appropriate, and (3) a reasonable guess can be made for a minimum and maximum, then the PERT distribution is a reasonable choice.

The commuter says that the “central” most likely time is 60 minutes, the minimum driving time is 50 minutes and under non-extreme events circumstances, maximum driving time would be 130 minutes. We can build a PERT distribution. It should look like this once it has been created.

Several things can be stated for this PERT. Its mean value is 70 and differs from the most likely or mode provided of 60. This difference between the mode and the mean arises from the fact that the distribution is assymmetric. The median also, is different to these two at approximately 68. The fact that extreme situations on the right hand side tail (extreme events of highway delays, infrequent flat-tires or accidents) extend the tail of the distribution towards the right-hand side. Therefore, the mean and the median become somehow "dragged" towards the right hand side of the curve. Thus, skewness is positive at 0.76.

The probability of a value larger or equal to 95 minutes is of 5%. In other words, one out of twenty possible outcomes (once a month on commuting days), the driver arrives on 95 minutes or more.

The driver could deem that this distribution is pretty punishing. It is assigning a lot of weight towards the right side of the distribution. Believe me, the TRIANGULAR does an even lousier job towards the right tail, but we will discuss this on another article.

That is why we propose here an ALTERNATE PERT. This is another sub-type of distribution provided by @RISK that does a much better job at assigning weight towards one tail of the distribution.

THE ALTERNATE PERT DISTRIBUTION

Considering "arriving late" as a critical value equal to 90 minutes, and that occurring once every 2 months, build an ALTPERT distribution with Min = 50, Mode = 60.

Start by inserting a PERT distribution.

2.      On the left hand side of the window where Parameters window shows Standard, click on the Drop Down option for the next Pert Parameters window to appear.

 3.      Here, click on the last option of Alternate Parameters.

4.      When clicking on Alternate Parameters, this same window extends itself allowing to change the standard parameters of a PERT (min, most likely and max) for as much as three alternate percentiles.  Since we are keeping the minimum and the most likely as they are, click on the Parameter Selection section to keep them checked. Leave the last percentile option checked and change the percentile value to 97.5%. Assuming the commuter goes to work some 20 times a month, then an event that happens every 2 months means that its frequency is of 1/40th. (2.5%) This fraction assumes that 97.5% of the time, he/she will arrive before 90 minutes.

5. Click on the Assign Excel Reference as Arguments button on the left hand side of the window beneath the Parameters section in order to retrieve the following window:

6.      Click on respectively B3, B4 and B15 cells to insert parameter references for the Minimum, Most Likely and the 97.5 percentile parameters.

7.      When done, press on the upper right hand-side corner of this small window to return to @RISK’s window, which will now look like this:

After clicking OK, the distribution window allows to insert the value 90 as the 97.5th percentile, thus creating a distribution at which one out of every 40 days, a value larger than 90 occurs.

8.      Edit the cell and change the hard code 97.5% value for a reference to cell B16. This will allow the model to be dynamically modified an easier way.

The syntax of this cell should look like the following once the hard coded references on the function have been changed to the respective cells: =RiskPertAlt("min",B3,"m. likely",B4,B16,B5)

Observe that @RISK allows for an alternative syntax in order to incorporate standard, alternate (percentiles) or hybrid parameter settings, such as this case.

The beauty of the Critical Value algorithm is that it focuses on one tail of the distribution, and not necessarily on its overall shape. We are not necessarily interested on exactly defining an absolute minimum AND and absolute maximum. We are interested on a point beyond which the analyzed phenomenom assumes a critical value or more.

A Critical Value could be any kind of threshold level: a budget, a physical or material constraint, a dead-line, a negotiated floor or ceiling level, a tolerance. It is easier for the expert to consider an extreme critical value than to focus on absolute minima and maxima.

This is why we strongly recommend the alternate PERT as opposed to a regular PERT or even a TRIANGULAR distribution.