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Triangular distribution – Strategy @ Risk

Strategy @ Risk

Tag: Triangular distribution

Project Management under Uncertainty
You can’t manage what you can’t measure
You can’t measure what you can’t define
How do you define something that isn’t known?

DeMarco, 1982

1.     Introduction

By the term Project we usually understand a unique, one-time operation designed to accomplish a set of objectives in a limited time frame. This could be building a new production plant, designing a new product or develop new software for a specific purpose.

A project usually differ from normal operations by; being a onetime operation, having a limited time horizon and budget, having unique specifications and by working across organizational boundaries. A project can be divides into four phases: project definition, planning, implementation and project phase-out.

2.     Project Scheduling

The project planning phase, which we will touch upon in this paper, consists of braking down the project into tasks that must be accomplished for the project to be finished.

The objectives of the project scheduling are to determine the earliest start and finish of each task in the project. The aim is to be able to complete the project as early as possible and to calculate the likelihood that the project will be completed within a certain time frame.

The dependencies[i] between the tasks determine their predecessor(s) and successor(s) and thus their sequence (order of execution) in the project[1]. The aim is to list all tasks (project activities), their sequence and duration[2] (estimated activity time length). The figure[ii] below shows a simple project network diagram, and we will in the following use this as an example[iii].

This project thus consists of a linear flow of coordinated tasks where in fact time, cost and performance can vary randomly.

A convenient way of organizing this information is by using a Gantt[iv] chart. This gives a graphic representation of the project’s tasks, the expected time it takes to complete them, and the sequence in which they must be done.

There will usually be more than one path (sequence of tasks) from the first to the last task in a project. The path that takes the longest time to complete is the projects critical path. The objective of all this is to identify this path and the time it takes to complete it.

3.     Critical Path Analysis

The Critical Path (CP)[v] is defined as the sequence of tasks that, if delayed – regardless of whether the other project tasks are completed on or before time – would delay the entire project.

The critical path is hence based on the forecasted duration of each task in the project. These durations are given as single point estimates[3] implying that the project’s tasks duration contain no uncertainty (deterministic). This is obviously wrong and will often lead to unrealistic project estimates due to the inherent uncertainty in all project work.

Have in mind that: All plans are estimates and are only as good as the task estimates.

As a matter of fact many different types of uncertainty can be expected in most projects:
1. Ordinary uncertainty, where time, cost and performance can vary randomly, but inside predictable ranges. Variations in task durations will cause the projects critical path to shift, but this can be predicted and the variation in total project time can be calculated.
2. Foreseen uncertainty, where a few known factors (events) can affect the project but in an unpredictable way[4]. This is projects where tasks and events occur probabilistic and contain logical relationships of a more complicated nature. E.g. from a specific event some tasks are undertaken with certainty while others probabilistically (Elmaghraby, 1964) and (Pritsker, 1966). The distribution for total project time can still be calculated, but will include variation from the chance events.
3. Unforeseen uncertainty, where one or more factors (events) cannot be predicted. This will imply that decisions points about the projects implementation have to be included at one or more points in the projects execution.
As a remedy to the critical path analysis inadequacy to the existence of ordinary uncertainty, the Program Evaluation and Review Technique (PERT[vi]) analysis was developed. PERT is a variation on Critical Path Analysis that takes a slightly more skeptical view of the duration estimates made for each of the project tasks.

PERT uses a tree-point estimate,[vii] based on the forecast of the shortest possible task duration, the most likely task duration and the worst-case task duration. The tasks expected duration is then calculated as a weighted average of these tree estimates of the durations.

This is assumed to help to bias time estimates away from the unrealistically short time-scales that often is the case.

4. CP, PERT and Monte Carlo Simulation

The two most important questions we want answered are:
- How long will it take to do the project?
- How likely is the project to succeed within the allotted time frame?
- In this example the projects time frame is set to 67 weeks.
We will use the Critical Path method, PERT and Monte Carlo simulation to try to answer these questions, but first we need to make some assumptions on the variability of the estimated task durations. We will assume that the durations are triangular distributed and that the actual durations can be both higher and lower than their most likely value.

The distributions will probably have a right tail since underestimation is common when assessing time and cost (positively skewed), but sometime people deliberately overestimate to avoid being responsible for later project delay (negatively skewed). The assumptions of the tasks duration are given in the table below:

The corresponding paths, critical path and project durations is given in the table below. The critical path method finds path #1 (tasks: A,B,C,D,E) as the critical path and thus expected project duration to 65 weeks. The second question however cannot be answered by using this method. So, in regard to probable deviations from expected project duration, the project manager is left without any information.

By using PERT, calculating expected durations and their standard deviation as described in endnote vii, we find the same critical path and roughly the same expected project duration (65.5 weeks), but since we now can calculate the estimate’s standard deviation we can find the probability of the project being finished inside the projects time frame.

By assuming that the sum of task durations along the critical path is approximately normal distributed, we find that the probability of having the project finished inside the time frame of 67 weeks to 79%. Since this gives is a fairly high probability of project success the manager can rest contentedly – or can she?

If we repeat the exercise, but now using Monte Carlo simulation we find a different answer. We can no longer with certainty establish a critical path. The tasks variability can in fact give three different critical paths. The most likely is path #1 as before, but there is a close to 30% probability that path #4 (tasks: A,B,C,G,E) will be the critical path. It is also possible, even if the probability is small (<5%), that path #3 (tasks: A,F,G,E) is the critical path (see figure below).So, in this case we cannot use the critical path method, it will give wrong answers and misleading information to the project manager and. More important is the fact that the method cannot use all the information we have about the project’s tasks, that is to say their variability.

A better approach is to simulate project time to find the distribution for total project duration. This distribution will then include the duration of all critical paths that may arise during the project simulation, given by the red curve in figure below:

This figure gives the cumulative probability distribution for the possible critical paths duration (Path#: 1,3,4) as well as for total project duration. Since path #1 consistently have long duration times there are only in ‘extreme’ cases that path #4 is the critical path. Most strikingly is the large variation in path #3’s duration and the fact that it can end up in some of the simulation’s runs as critical path.

The only way to find the distribution for total project duration is for every run in the simulation to find the critical path and calculate its duration.

We now find the expected total project duration to be 67 weeks, one week more than what the CPM and PERT gave, but more important, we find that the probability of finishing the project inside the time frame is only 50%.

By neglecting the probability that the critical path might change due to task variability PERT is underestimating project variance and thus the probability that the project will not finish inside the expected time frame.

Monte Carlo models like this can be extended to include many types of uncertainty belonging to the classes of foreseen and unforeseen uncertainty. However, it will only be complete when all types of project costs and their variability are included.

5. Summary

Key findings in comparative studies show that using Monte Carlo along with project planning techniques allows better understanding of project uncertainty and its risk level as well as provides project team with the ability to grasp various possible courses of the project within one simulation procedure.

Notes

[1] This can be visualized in a Precedence Diagram also known as a Project Network Diagram.In a Network Diagram, the start of an activity must be linked to the end of another activity

[2] An event or a milestone is a point in time having no duration. A Precedence Diagram will always have a Start and an End event.

[3] As a “best guess” or “best estimate” of a fixed or random variable.

[4] E.g. repetition of tasks.

Endnotes

[i] There are four types of dependencies in a Precedence Diagram:
1. Finish-Start: A task cannot start before a previous task has ended.
2. Start-Start: There is a defined relationship between the start of tasks.
3. Finish-Finish: There is a defined relationship between the end dates of tasks.
4. Start-Finish: There is a defined relationship between the start of one task and the end date of a successor task.
[ii] Taken from the Wikipedia article: Critical path drag, http://en.wikipedia.org/wiki/Critical_path_drag

[iii] The Diagram contains more information than we will use. The diagram is mostly self-explaining, however Float (or Slack) and Drag is defined as the activity delay that the project can tolerate before the project comes in late and how much a task on the critical path is delaying project completion (Devaux,2012).

[iv] The Gantt chart was developed by Henry Laurence Gantt in the 1910s.

[v] The Critical Path Method (CPM) was developed in the late 1950s by Morgan R. Walker of DuPont and James E. Kelley, Jr. of Remington Rand.

[vi] The Program Evaluation and Review Technique (PERT) were developed by Booz Allen Hamilton and the U.S. Navy, at about the same time as the CPM. Key features of a PERT network are:
1. Events must take place in a logical order.
2. Activities represent the time and the work it takes to get from one event to another.
3. No event can be considered reached until ALL activities leading to the event are completed.
4. No activity may be begun until the event preceding it has been reached.
[vii] Assuming, that a process with a double-triangular distribution underlies the actual task durations, the tree estimated values (min, ml, max) can then be used to calculate expected value (E) and standard deviation (SD) as L-estimators, with: E = (min + 4m + max)/6 and SD = (max − min)/6.

E is thus a weighted average, taking into account both the most optimistic and most pessimistic estimates of the durations provided. SD measures the variability or uncertainty in the estimated durations.

References

Devaux, Stephen A.,(2012). “The Drag Efficient: The Missing Quantification of Time on the Critical Path” Defense AT&L magazine of the Defense Acquisition University. Retrieved from http://www.dau.mil/pubscats/ATL%20Docs/Jan_Feb_2012/Devaux.pdf

DeMarco, T, (1982), Controlling Software Projects, Prentice-Hall, Englewood Cliffs, N.J., 1982

Elmaghraby, S.E., (1964), An algebra for the Analyses of Generalized Activity Networks, Management Science, 10,3.

Pritsker, A. A. B. (1966). GERT: Graphical Evaluation and Review Technique (PDF). The RAND Corporation, RM-4973-NASA.
01/08/2014
Inventory management – Stochastic supply
This entry is part 4 of 4 in the series Predictive Analytics

Introduction

We will now return to the newsvendor who was facing a onetime purchasing decision; where to set the inventory level to maximize expected profit – given his knowledge of the demand distribution. It turned out that even if we did not know the closed form (( In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain “well-known” functions.)) of the demand distribution, we could find the inventory level that maximized profit and how this affected the vendor’s risk – assuming that his supply with certainty could be fixed to that level. But what if that is not the case? What if the supply his supply is uncertain? Can we still optimize his inventory level?

We will look at to slightly different cases:
1. one where supply is uniformly distributed, with actual delivery from 80% to 100% of his ordered volume and
2. the other where the supply have a triangular distribution, with actual delivery from 80% to 105% of his ordered volume, but with most likely delivery at 100%.
The demand distribution is as shown below (as before):

Maximizing profit – uniformly distributed supply

The figure below indicates what happens as we change the inventory level – given fixed supply (blue line). We can see as we successively move to higher inventory levels (from left to right on the x-axis) that expected profit will increase to a point of maximum.

If we let the actual delivery follow the uniform distribution described above, and successively changes the order point expected profit will follow the red line in the graph below. We can see that the new order point is to the right and further out on the inventory axis (order point). The vendor is forced to order more newspapers to ‘outweigh’ the supply uncertainty:

At the point of maximum profit the actual deliveries spans from 2300 to 2900 units with a mean close to the inventory level giving maximum profit for the fixed supply case:

The realized profits are as shown in the frequency graph below:

Average profit has to some extent been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation, this variability has increased by almost 13%, and this is mainly caused by an increased negative skewness – the down side has been raised.

Maximizing profit – triangular distributed supply

Again we compare the expected profit with delivery following the triangular distribution as described above (red line) with the expected profit created by known and fixed supply (blue line). We can see as we successively move to higher inventory levels (from left to right on the x-axis) that expected profits will increase to a point of maximum. However the order point for the stochastic supply is to the right and further out on the inventory axis than for the non-stochastic case:

The uncertain supply again forces the vendor to order more newspapers to ‘outweigh’ the supply uncertainty:

At the point of maximum profit the actual deliveries spans from 2250 to 2900 units with a mean again close to the inventory level giving maximum profit for the fixed supply case ((This is not necessarily true for other combinations of demand and supply distributions.)) .

The realized profits are as shown in the frequency graph below:

Average profit has somewhat been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation this variability has increased by 10%, and this is again mainly caused by an increased negative skewness – again have the down side been raised.

The introduction of uncertain supply has shown that profit can still be maximized however the profit will be reduced by increased costs both in lost sales and in excess inventory. But most important, profit variability will increase raising issues of possible other strategies.

Summary

We have shown through Monte-Carlo simulations, that the ‘order point’ when the actual delivered amount is uncertain can be calculated without knowing the closed form of the demand distribution. We actually do not need the closed form for the distribution describing delivery, only historic data for the supplier’s performance (reliability).

Since we do not need the closed form of the demand distribution or supply, we are not limited to such distributions, but can use historic data to describe the uncertainty as frequency distributions. Expanding the scope of analysis to include supply disruptions, localization of inventory etc. is thus a natural extension of this method.

This opens for use of robust and efficient methods and techniques for solving problems in inventory management unrestricted by the form of the demand distribution and best of all, the results given as graphs will be more easily communicated to all parties than pure mathematical descriptions of the solutions.

Average profit has to some extent been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation, this variability has increased by almost 13%, and this is mainly caused by an increased negative skewness – the down side has been raised.
23/09/2012
Plans based on average assumptions ……
This entry is part 3 of 4 in the series The fallacies of scenario analysis

The Flaw of Averages states that: Plans based on the assumption that average conditions will occur are usually wrong. (Savage, 2002)

Many economists use what they believe to be most likely ((Most likely estimates are often made in-house based on experience and knowledge about their operations.)) or average values ((Forecasts for many types of variable can be bought from suppliers of ‘consensus forecasts’.)) (Timmermann, 2006) (Gavin & Pande, 2008) as input for the exogenous variables in their spreadsheet calculations.

We know however that:
1. the probability for any variable to have outcomes equal to any of these values is close to zero,
1. and that the probability of having outcomes for all the (exogenous) variables in the spreadsheet model equal to their average is virtually zero.
So why do they do it? They obviously lack the necessary tools to calculate with uncertainty!

But if a small deviation from the most likely value is admissible, how often will the use of a single estimate like the most probable value be ‘correct’?

We can try to answer that by looking at some probability distributions that may represent the ‘mechanism’ generating some of these variables.

Let’s assume that we are entering into a market with a new product, we know of course the maximum upper and lower limit of our future possible market share, but not the actual number so we guess it to be the average value = 0,5. Since we have no prior knowledge we have to assume that the market share is uniformly distributed between zero and one:

If we then plan sales and production for a market share between 0, 4 and 0, 5 – we would out of a hundred trials only have guessed the market share correctly 13 times. In fact we would have overestimated the market share 31 times and underestimated it 56 times.

Let’s assume a production process where the acceptable deviation from some fixed measurement is 0,5 mm and where the actual deviation have a normal distribution with expected deviation equal to zero, but with a standard deviation of one:

Using the average deviation to calculate the expected error rate will falsely lead to us to believe it to be zero, while it in fact in the long run will be 64 %.

Let’s assume that we have a contract for drilling a tunnel, and that the cost will depend on the hardness of the rock to be drilled. The contract states that we will have to pay a minimum of $ 0.5M and a maximum of $ 2M, with the most likely cost being $ 1M. The contract and our imperfect knowledge of the geology make us assume the cost distribution to be triangular:

Using the average ((The bin containing the average in the histogram.)) as an estimate for expected cost will give a correct answer in only 14 out of a 100 trials, with cost being lower in 45 and higher in 41.

Now, let’s assume that we are performing deep sea drilling for oil and that we have a single estimate for the cost to be $ 500M. However we expect the cost deviation to be distributed as in the figure below, with a typical small negative cost deviation and on average a small positive deviation:

So, for all practical purposes this is considered as a low economic risk operation. What they have failed to do is to look at the tails of the cost deviation distribution that turns out to be Cauchy distributed with long tails, including the possibility of catastrophic events:

The event far out on the right tail might be considered a Black Swan (Taleb, 2007), but as we now know they happen from time to time.

So even more important than the fact that using a single estimate will prove you wrong most of the times it will also obscure what you do not know – the risk of being wrong.

Don’t worry about the average, worry about how large the variations are, how frequent they occur and why they exists. (Fung, 2010)

Rather than “Give me a number for my report,” what every executive should be saying is “Give me a distribution for my simulation.”(Savage, 2002)

References

Gavin,W.,T. & Pande,G.(2008), FOMC Consensus Forecasts, Federal Reserve Bank of St. Louis Review, May/June 2008, 90(3, Part 1), pp. 149-63.

Fung, K., (2010). Numbers Rule Your World. New York: McGraw-Hill.

Savage, L., S.,(2002). The Flaw of Averages. Harvard Business Review, (November), 20-21.

Savage, L., S., & Danziger, J. (2009). The Flaw of Averages. New York: Wiley.

Taleb, N., (2007). The Black Swan. New York: Random House.

Timmermann, A.,(2006). An Evaluation of the World Economic Outlook Forecasts, IMF Working Paper WP/06/59, www.imf.org/external/pubs/ft/wp/2006/wp0659.pdf

Endnotes
01/03/2011
Corporate Risk Analysis

This entry is part 2 of 6 in the series Balance simulation

Strategy @Risk has developed a radical and new approach to the way risk is assessed and measured when considering current and future investment. A key part of our activity in this sensitive arena has been the development of a series of financial models that facilitate understanding and measurement of risk set against a variety of operating scenarios.

We have written a paper which outlines our approach to Corporate Risk Analysis to outline our approach. Read it here.

Our purpose in this paper is to show that every item written into a firm’s profit and loss account and its balance sheet is a stochastic variable with a probability distribution derived from probability distributions for each factor of production. Using this approach we are able to derive a probability distribution for any measure used in valuing companies and in evaluating strategic investment decisions. Indeed, using this evaluation approach we are able to calculate expected gain, loss and probability when investing in a company where the capitalized value (price) is known.

03/02/2009
Uncertainty – lack of information
This entry is part 3 of 6 in the series Monte Carlo Simulation

Every item in a budget or a profit and loss account represents in reality a probability distribution. In this framework all items whether from the profit and loss account or from the balance sheet will have individual probability distributions. These distributions are generated by the combination of distributions from factors of production that define the item.

Variance will increase as we move down the items in the profit and loss account. The message is that even if there is a low variance in the input variables (sales, prices, costs etc.) metrics like NOPLAT, Free Cash Flow and Economic Profit will have a much higher variance.

The key issue is to identify the various items and establish the individual probability distribution. This can take place by using historical data, interviewing experts or comparing data from other relevant sources. There are three questions we need to answer to define the proportions of the uncertainty:
- What is the expected value?
- What is the lowest likely value?
- What is the highest likely value?
When we have decided the limits where we with 95% probability estimate the result to be within we then decide what kind of probability distribution is relevant for the item. There are several to choose among, but we will emphasize three types here.
1. The Normal Distribution
2. The Skewed Distribution
3. The Triangular Distribution
The Normal Distribution is being used when we have situations where there is a likeliness for a symmetric result. It can be a good result but has the same probability of being bad.

The Skew Distribution is being used when it can occur situations where we are lucky and experience more sales than we expected and vice versa we can experience situations where expenditure is less than expected.

The Triangular Distribution is being used when we are planning investments. This is due to the fact that we tend to know fairly well what we expect to pay and we know we will not get merchandise for free and there is a limit for how much we are willing to pay.

When we have defined the limits for the uncertainty where we with 95% probability estimate the result to be within we can start to calculate the risk and prioritize the items that matters in terms of creating value or loss.
01/09/2008