My recent blogs discussed how to determine an Economic Lot Size to balance obsolescence, inventory, and changeover costs for perishable products when demand is uncertain. I recommended using a Gamma distribution to model demand during the shippable life. A Gamma distribution might also be used as basis for Statistical Safety stock calculations.
Using a Gamma distribution to project demand variability requires a reliable basis for estimating the distribution. This would usually be based on several years of historical data. However, in times like this current pandemic, history is unreliable as a predictor of future demand. This blog will propose methods for estimating the Gamma distribution function when history is not valid.
I would like to borrow a concept from the Project Management Body of Knowledge (PMBOK), from the Project Management Institute. There are a few different ways it might be applied.
Getting Estimates
All the methods described below start with getting estimates from your Subject Matter Experts (SMEs). Since we are estimating demand, these SMEs are likely to be your sales team, your marketing team, your customers, or some combination thereof. If there are several SMEs that are knowledgeable about a given product, one might poll them all and average the results.
In any of the approaches described, one always asks for a Best Guess estimate. Depending on which approach you use, you will also need either a high estimate, a low estimate, or both. If asking for all three, consider the best way to get estimates. People often have a cognitive bias to center the best guess between the high and low estimates. In some circumstances, this may not be valid. For this reason, it may be best not to get all three estimates from the same person at the same time.
Method 1 – The PERT Distribution
PMBOK includes the PERT distribution as a method to estimate the duration for a task when there is no previous history of similar projects to fall back on. The PERT distribution is a transformation of the four-parameter Beta distribution. The mode of the PERT distribution matches the best guess estimate and there is zero probability of values less than the low estimate or greater than the high estimate.
The PERT distribution requires three estimates – low estimate (l), best guess (b), and high estimate (h). The best guess estimate is weighted four times and the other two weighted one times in calculating the estimated mean. The estimated mean is calculated as (l +4b + h) / 6. The estimated standard deviation is (h – l) / 6.
Calculations for the PERT Distribution
In all formulas below, the following symbols are used:
- l = Low estimate (minimum value) of the distribution;
- h = High estimate (maximum value) of the distribution;
- b = Best guess (which with the PERT distribution is the mode) of the distribution; and;
- α and β are shape parameters of the PERT distribution.
The probability density function of PERT distribution is given as:
Where 
is the Beta Function, which is
The PERT distribution is defined so that the mean and standard deviation take the following values:
For the distribution to have this mean and standard deviation, α and β are calculated as follows:
Strengths and Weaknesses of the PERT Distribution
The PERT distribution is designed to force the mode to match the Best Guess estimate. In this approach, the best guess might even be better called the most likely estimate.
The PERT distribution does not allow any values greater than the upper estimate or less than the lower estimate. If you are certain that the high estimate is an upper bound and the low estimate is a lower bound, then the PERT distribution might be the better fit. If you believe there could be values greater than the high estimate or less than the low estimate, then an approach that uses a Gamma distribution might be a better option.
This approach is not valid if there is a significant probability of zero demand.
Method 2 – The Gamma Distribution Using PERT Estimates
I propose using the PERT method as an approximation to the Gamma distribution for demand during a specified lead time. In this approach, one might start with same three estimates – low estimate (l), best guess (b), and high estimate (h). Each of these estimates may be weighted as desired to calculate a mean. To match the PERT method most closely, the estimated mean is calculated as (l +4b + h) / 6. The estimated standard deviation is (h – l) / 6. These values for mean and standard deviation can then be used to generate a Gamma Distribution.
To see the formulas for a Gamma distribution, see my previous blog, Estimating Expected Obsolescence-for a Given Starting Inventory.
Strengths and Weaknesses of the Gamma Distribution Using PERT Estimates
Generally, with this method neither the mode nor the mean of the Gamma distribution will align with the best guess estimate, which is not intuitive. The mode of the Gamma distribution is always slightly below the mean, but neither will align with the best guess.
This method yields a distribution that has a small probability of values less than the low estimate or greater than the high estimate. This is intuitively reasonable given that we are estimating uncertain demand.
Calculating the Gamma distribution this way yields a greater probability that the result will be close to the mean. If greater variability is predicted, the curve can be flattened by increasing the standard deviation. For example, if you estimate there is only a 90% probability of being between the low and high estimates, you might calculate the estimated standard deviation as (h – l) / 3. This would give a flatter distribution but would also give a greater probability of values not between the low and high estimates
This approach is not valid if there is a significant probability of zero demand.
Comparison of PERT and Gamma Distributions
Figure 1 shows examples of Gamma distributions and PERT distributions for various values of mean and standard deviation. The top left shows an example where the best guess estimate is centered between the high estimate and low estimate. In this example both the Gamma distribution and PERT distribution approximate a Normal distribution.
In the bottom left example, the low estimate is significantly further from the best guess. Note how both the Gamma distribution and PERT distribution have extremely low probabilities before they get near the low estimate. Note how the mode of the PERT distribution is exactly at the best guess, but the mode of the Gamma distribution is lower.
In the right two examples the best guess is quite close to the low estimate or the high estimate. Note how the PERT distribution has zero probability of being less than the low estimate or more than the high estimate. However, in these scenarios the Gamma distribution has a noticeable probability of values less than the low estimate or more than the high estimate.
The PERT approach of three estimates can be used to generate either a PERT distribution or a Gamma distribution with similar mean and variance. I generally recommend using the Gamma distribution to estimate demand, and there may be a better option for estimating the Gamma distribution.
Method 3 – The Gamma Distribution Using Two Estimates
It’s also possible to fit a Gamma distribution so that the mode matches the best estimate. In this case, it may not be of value to also ask for both a high estimate and a low estimate. One might ask for just best guess and a high estimate, or just best guess and a low estimate. The best guess is used as the mode of the Gamma distribution and the difference between best guess and low or high is used to estimate the standard deviation.
Calculations for the Gamma Distribution Using Two Estimates
In all formulas below, the following symbols are used:
- b = Best Guess (Mode) of the distribution;
- h = High estimate of the distribution;
- σ = Standard deviation of the distribution; and;
- α and β are scale and shape parameters of the Gamma distribution.
The mode of a Gamma distribution is (α-1)β and the standard deviation is . The probability density and other calculations for the Gamma distribution are given in my previous blog, Estimating Expected Obsolescence-for a Given Starting Inventory.
Setting the mode to equal the best guess and solving for α:
Solving the standard deviation formula for β
Substituting from formula above for α
Solving the quadratic equation (the negative root is ignored)
And from above,
If you are 99% certain that demand will not be greater than the high estimate, you might estimate the standard deviation as
The calculations for using a low estimate (l) instead of a high estimate (h) are the same except the standard deviation is
As discussed above, the denominator could be reduced to get a wider distribution, if desired.
Figure 2 shows four examples of this method for different best guess estimates.
Strengths and Weaknesses of the Gamma Distribution Using Two Estimates
Like the PERT distribution, this method yields a distribution whose mode matches the best guess estimate. Once again, the best guess might even be better called the most likely estimate, as we are forcing the mode to match the best guess.
There is a small probability of values less than the low estimate or greater than the high estimate. This is intuitively reasonable given that we are estimating uncertain demand. However, this approach is not valid if there is a significant probability of zero demand. A different approach is needed in that scenario.
Method 4 – When there is a Significant Probability of Zero Demand.
What if there is a significant probability of no demand? You might have a SME that says something like there is 40% chance of no demand during the period in question, but the best guess is some number of units.
Both the Gamma and Beta distributions have zero probability of no demand. One might try to find parameters that give a distribution with say a 30% probability of demand less than 100 units. However, forcing these parameters in most cases forces a mode close to zero. So there would not be a mode at the best guess value.
Rather than try to force a distribution that has some percentage probability of near-zero demand, it might be more effective to combine a probability of zero demand with one of above methods. One might say, for example, that there is 30% probability of no demand, but if there is demand it will be distributed as a Gamma distribution with mode (best guess) at 500 units. The expected value of demand would then be 70% of the mean value of the distribution. Formulas for calculating safety stock or estimating obsolescence could be adjusted for this probability.
Conclusion
If future demand is not expected to follow history, you will need to rely on input from SMEs to develop any estimate for future demand. Before deciding on one of the above methods, it might be helpful to get SME input on key questions such as,
- Is there an absolute maximum value that demand cannot exceed?
- Is there an absolute minimum value that demand cannot go below?
- How close to the best guess estimate do you expect demand to cluster?
- What is the probability that there could be no demand at all during the projected period?
Which of above methods you choose to use will depend on the answers to this question and your intuitive understanding of the likely demand patterns. The mean and standard deviation for demand can then be estimated from two or three estimates as described above. No matter which method you use, the value of any of these approaches is limited by the quality of the estimates provided by your Subject Matter Experts.
If the Gamma distribution is determined to be an acceptable approximation of demand variability, the resulting values for mean and standard deviation can be used in the formulas in my prior blog to calculate Economic Lot Size with uncertain demand. If a PERT distribution or another distribution is determined to be a better fit, the formulas in my prior blog will need to be adapted to reflect the different probability distribution.
References for More Information on PERT and Gamma Distributions
Three Point Estimating and PERT Distribution Project-Management.Info
Peter J. Sherman Better Project Management Through Beta Distribution isixsigma.com
Statistics – Beta Distribution TutorialsPoint.com
T. A. Burgin, 1975. The Gamma Distribution and Inventory Control Operational Research Quarterly (1970-1977), Vol. 26, No. 3, Part 1 (Sep., 1975), pp.507-525
R. D. Snyder, 1984. Inventory control with the gamma probability distribution European Journal of Operational Research Volume 17, Issue 3, September 1984, Pages 373-381
Ashkan Mirzaee, 2017. Alternative Methods for Calculating Optimal Safety Stock Levels