Project cost risk analysis

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A cost risk analysis is usually developed from a work breakdown structure (WBS) which is a document that details, from the top down, the various work packages (WPs) of which the project consists. Each WP may then be subdivided into a bill of quantities and estimates of the labour required to complete them.

There will usually be a number of cost items associated with each WP that have an element of uncertainty. In addition, there may be discrete events (risks or opportunities) that could change the size of these costs. The normal uncertainties in the cost items are modelled by continuous distributions like the PERT or Triangle: we will use the Triangle distribution for the rest of this explanation for simplicity but the reader should be aware of the drawbacks of this distribution. The impact of the risks and opportunities will similarly be modelled by continuous distributions but whether they occur, or not, is modelled with a Discrete distribution. To illustrate this, consider the following example.

Example 1

A new office block has been designed to be roofed with corrugated galvanised steel at a cost of between £165 000 and £173 000, but most probably £167 000. However, the council's planning department has received a number of objections from local residents. The architect thinks there is about a 30% chance that the building will have to be roofed with slate at a cost of between £193 000 and £208 000, but most likely about £198 000.

The example model below shows how to model the roofing's cost using Triangle distributions. The model selects a steel roof (Cell C5) for 70% of the scenarios and a slate roof (Cell C6) for the remaining 30% to produce a combined uncertainty in Cell C8.

Example model Roof_of_office_block - cost of the roofing using Triangle distributions

Many of the cost items for the project will be in the form of: x items at £y/item where x and y are both uncertain quantities. At first sight, it seems logical to simply multiply these two variables together to get the cost, i.e. cost = x · y. However, there is a potential problem with this approach (determining the sum of random variables is discussed in detail here). Consider the following example:

Example 2

A ship's hull consists of 562 plates, each of which has to be riveted in place. Each plate is riveted by one worker. The supervisor, reflecting on the efficiency of her work force, considers that the best her riveters have ever done is put up a plate in 3 h 45 min. On the other hand, the longest they have taken is about 5 h 30 min, and it is far more likely that a plate would be riveted in about 4 h 15 min. Each riveter is paid £7.50 an hour. What is the total labour cost for riveting? One's first thought might be to model the total cost as follows:

Cost = 562 * Triangle(3.75, 4.25, 5.5) * £7.50

What happens if we run a simulation on this formula? In some iterations, we will produce values close to 3.75 from the Triangle distribution. This is saying that all of the plates could have been riveted in record time - clearly not a realistic scenario. Similarly, some iterations will generate values close to 5.5 from the Triangle distribution; the scenario that the work force took as long to put up every plate on the ship as it took them to do the trickiest plate in memory.

The problem lies in the fact that the Triangle distribution is modelling the uncertainty of an individual plate but we are using it as if it were the distribution of the average time for 562 plates.

There are essentially two approaches to the correct modelling of this problem. The first is to model each plate separately, i.e. set up a column of 562 Triangle(3.75, 4.25, 5.5) distributions, add them up and multiply the sum by £7.50. Whilst this is quite correct, it is obviously impractical to use a spreadsheet model of 562 cells just for this one cost item, so the technique is only really useful if there are just a few items to be summed.

The second option is to apply the Central Limit theorem. The mean m and standard deviation s of a Triangle(3.75, 4.25, 5.5) distribution are:

Since there are 562 items, the distribution of the total person hours for the job is given by

Then, the total labour cost for riveting is estimated as:

Once each cost item for the project has been identified, along with any associated risks and uncertainties, a model can be produced to estimate the total project cost. The example model Cost_model illustrates the sort of model structure that could be used. In this example, each item is clearly defined along with the assumptions that are used in its estimation and the impacts and probabilities of any risks.

One question that management will often ask is how the budget and contingency are distributed back among the cost items. This knowledge will help the project manager to keep an eye on how the project is progressing. Our approach is to distribute back the budget and contingency costs so that the figures associated with each cost item have the same probability of being exceeded, as explained here.

Read on: Schedule risk analysis