The basic principle of cost estimation is deceptively simple. To estimate the cost for each item in a project, multiply the quantity of work $Q_i$ times the rate $R_i$ you will pay for each unit of work. Sum the cost of each item, and you have the total project cost $P$ for a project with $j$ items:

$$P = \sum_{i=1}^j Q_i R_i$$

The reality is, unfortunately, a bit more inconsistent than this equation can express. Determining the correct quantity and the rate for each item is a fine art that requires knowledge of both engineering and economics.

Cost estimates reflect the state of knowledge at the time of developing them. Many aspects of the project are not known and might only surface once we start digging. As the late Donald Rumsfeld philosophically said:

… there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say, we know that there are some things that we do not know. But there are also unknown unknowns — the ones we don’t know, we don’t know.

The probabilistic approach in the table above is an acceptable way to assign uncertainties and risks, but it does not give us any insight into how likely this estimate will eventuate.

A Monte Carlo Simulation can further refine the cost estimate by digging deeper into the uncertainty. Monte Carlo simulations are a class of methods that use large volumes of randomised numbers within known distributions to simulate reality. They can be used for any situation where deterministic methods fail, such as modelling contact centre call traffic.

To add more information to the estimate, we assign a low, likely and high cost to each item. The low cost ($a$) will rarely be achieved and is your ‘bargain-basement’ assessment. The likely cost ($c$) is what you would typically use in your estimate. Finally, the high cost ($b$) is what you will pay when everything you can think of goes wrong. Statistically, each cost item now has a triangular probability distribution, visualised below.

Our cost estimate would now look something like this:

For each item we can calculate the average or any other percentile. The average cost of an item is $\frac{a+b+c}{3}$. The expected cost $x$ with likelihood $p$ is defined by the quantiles for each item, given by:

$$x_p = \begin{cases} a + \sqrt{(b-a)(c-a)p} & \text{for } 0 \leq p \leq F(c) \\ b – \sqrt{(b-a)(b-c)(1-p)} & \text{for } F(c) \leq p \leq 1 \end{cases}$$

Where $F(c)$ is the result of the cumulative distribution: $F(c) = \frac{(x-a)^2}{(b-a)(c-a)}$.

Note that these formulas only work when: $a \leq c$ and $b \geq c$.

While the triangular distribution is the most common method to assess uncertainty. Other distributions are possible, but the principles remain the same. Be mindful, however, that most probability distributions range from minus to plus infinity, which is not a realistic assumption for cost estimates.

If your cost estimate contains only one item, you are done because you can calculate the project cost by plugging a percentile into the formulas. But when the estimate includes multiple items, doing so analytically will be a bit too complex for mere mortals. This is where the Monte Carlo technique comes in.

Monte Carlo simulates reality by calculating thousands of possible outcomes of the distribution. Several libraries are available in R that can calculate triangular distributions. The triangle package provides a set of functions to work with triangular distributions. The rtriangle() function provides a vector of random quantiles for the chosen distribution, as shown below:

library(triangle)
hist(rtriangle(n = 10000, a = 12000, b = 15000, c = 14000), 
     breaks = 100, 
     main = "Triangular distribution simulation")

When plotting the histogram of these results, the triangular shape becomes evident, albeit a bit rough around the edges. Notice that a Monte Carlo simulation is only ever an estimate. The higher the number of simulations (in this diagram 10,000), the better the result. Change the n parameter in the rtriangle function to see for yourself.

To add the probability distribution of all your cost items, you can store all simulations in a matrix with n columns (number of simulations) and j (number of cost items) rows. Then, in the next step, you add all the simulated values and create a histogram or calculate percentiles.

The example below reads a CSV file with the same content as the table above. Then, it creates a results matrix, runs the simulations and calculates the result.

  ## Read Data
  estimate <- read.csv("estimate.csv")

  ## Simulation settings
  n <- 10000
  j <- nrow(estimate)
  mc_sims <- matrix(nrow = j,
                    ncol = n)

  ## Simulation
  for (i in 1:j){
      mc_sims[i,] <- rtriangle(n = it,
                               a = estimate$Low[i],
                               b = estimate$High[i],
                               c = estimate$Medium[i])}

  ## Determine estimates and 95th percentile
  mc_results <- colSums(mc_sims)
  p95 <- quantile(mc_results, 0.95)

  ## Visualise
  hist(mc_results, breaks = 100)
  abline(v = p95, col = "red", lwd = 2)

Download from GitHub

A Monte Carlo simulation outcome is thus never one number but a vector of numbers from which you can analyse. The example below uses the 95^th percentile as the budget figure.

The addition of triangular distributions will create a new distribution specific to this project. The mc_results vector holds the estimated possible total cost.

To set a budget figure, you need to nominate a percentile. The 50^th percentile has an equal chance of being higher or lower than the actual. To have some more certainty that a sufficient budget is available, perhaps you should choose the 95^th percentile, or whatever percentile matches your risk appetite.

Monte Carlo cost estimates are a powerful tool to better understand your project. They are, however, like any modelling, subject to the GiGo-principle (Garbage-in-Garbage-out). Therefore, determining the low, likely and high cost will require domain knowledge. While these three estimates themselves have their own level of uncertainty, they will provide better insights than relying only on the likely estimate.

Probabilistic cost estimates do not guarantee that your project will remain under budget, but they certainly help you achieve this elusive goal.