Travailler avec des futurs incertains
Le futur est incertain. Pourtant, presque toutes les solutions prédictives de supply chain font l’hypothèse inverse : elles supposent que leurs prévisions sont correctes, puis déroulent leurs simulations sur la base de ces prévisions. Implicitement, le futur est supposé certain et les complications s’ensuivent.
D’un point de vue historique, les ingénieurs logiciels ne faisaient pas ces hypothèses sans raison : un futur déterministe était la seule option que les premiers ordinateurs - et même les moins anciens - pouvaient traiter correctement. Ainsi, bien que gérer un futur incertain fût reconnu comme la meilleure approche en théorie, en pratique, ce n’était même pas une option.
En outre, quelques astuces mathématiques ont été découvertes dès le début du XXe siècle afin de contourner ce problème. Par exemple, l’analyse classique des stocks de sécurité suppose que les lead times et la demande suivent tous deux une distribution normale. L’hypothèse de la distribution normale est commode d’un point de vue informatique, car il suffit de deux variables pour modéliser le futur : la moyenne et la variance.
Yet again, the normal distribution assumption - both for the lead times and the demand - proved to be incorrect in nearly all but a few situations, and complications ensued.
Back in 2012 at Lokad, we realized that the classic inventory forecasting approach was simply not working: mean or median forecasts were not addressing the right problem. No matter how much technology we poured on the case, it was not going to work satisfyingly.
Thus, we shifted to quantile forecasts, which can be interpreted as forecasting the future with an intended bias. Soon we realized that quantiles were invariably superior to the classic safety stock analysis, if only because quantiles were zooming in on where it really mattered from a supply chain perspective.
However, while going quantile, we realized that we had lost quite a few things in the process. Indeed, unlike classic mean forecasts, quantile forecasts are not additive, so it was not possible to make sense of a sum of those quantiles for example. In practice, the loss wasn’t too great because since classic forecasts weren’t making much sense in the first place, summing them up wasn’t a reasonable option anyway.
Over the years, while working with quantiles, we realized that so many of the things we took for granted had become a lot more complicated: demand quantities could no longer be summed or subtracted or linearly adjusted. In short, while moving towards an uncertain future, we had lost the tools to operate on this uncertain future.
Back in 2015, we introduced quantile grids. While quantile grids were not exactly the same as our full-fledged probabilistic forecasts just yet, our forecasting engine was already starting to deliver probabilities instead of quantile estimates. Distributions of probabilities are much more expressive than simple quantile estimates, and, it turns out that it is possible to define an algebra over distributions.
While the term algebra might sound technical, it’s not that complicated; it means that a simple operation such as the sum, the product, the difference, …, can be defined in ways which are not only mathematically consistent, but also highly relevant from the supply chain perspective.
As a result, just a few weeks ago, we integrated an algebra of distributions right into Envision, our domain-specific language dedicated to commerce optimization. Thanks to this algebra of distributions, it becomes straightforward to carry out_seemingly_ simple operations such as summing two uncertain lead times (say an uncertain production lead time plus an uncertain transport lead time). The sum of those two lead times is carried out through an operation known as a convolution. While the calculation itself is fairly technical, in Envision, all it takes is to write A = B +* C, where +* is the convolution operator used to sum up independent random variables (*).
Through this algebra of distributions most of the “intuitive” operations which were possible with classic forecasts are back : random variables can be summed, multiplied, stretched, exponentiated, etc. And while relatively complex calculations are taking place behind the scenes, probabilistic formulas are not more complicated than plain Excel formulas from the Envision perspective.
Instead of wishing for the forecasts to be perfectly accurate, this algebra of distributions lets us embrace uncertain futures: supplier lead times tend to vary, quantities delivered may differ from quantities ordered, customer demand changes, products get returned, inventory may get lost or damaged … Through this algebra of distributions it becomes much more straightforward to model most of those uncertain events with minimal coding efforts.
Under the hood, processing distributions is quite intensive; and once again, we would never have ventured into those territories without a cloud computing platform that handles this type of workload - Microsoft Azure in our case. Nevertheless, computing resources have never been cheaper, and your company’s next $100k purchase order is probably well worth spending a few CPU hours - costing less than $1 and executed in just a few minutes - to make sure that the ordered quantities are sound.
(*) A random variable is a distribution that has a mass of 1. It’s a special type of distribution. Envision can process distributions of probabilities (aka random variables), but more general distributions as well.
