We live in the age of uncertainty: uncertainty causes your mortgage rate to fluctuate; your supermarket shelves to remain empty; and your energy price to skyrocket; but what is uncertainty? 

To shed light on the nature of uncertainty, we may consider the following question: is there such a thing as decision making under certainty? To investigate this question, we must then dissect this notion. 

'Certain' is an ambiguous word: it can mean 'sure' - as in "I am certain about this;" but it can also mean particular: "a certain person." 'Uncertain' does not have this ambiguity, 'unsure' being a close synonym. 

To fully grasp the meaning of a word, it is often helpful to consider its etymology. 'Certain' comes from 'certus,' the past participle of 'cernere,' which means 'to distinguish,' 'to decide,' literally 'to sift,' 'to separate.' The associated proto-indo-european root means 'to sieve,' 'to discriminate,' 'to distinguish.' The same root is also the source of ancient Greek 'krisis:' 'turning point,' 'judgment,' 'result of a trial;' from which the English word 'crisis' also originates, albeit with a slightly different meaning that relates to the trouble one experiences in the decision-making process. 

After analyzing the etymology of 'certain' let's now turn to the word 'sure.' 'Sure' comes from Latin 'securus,' literally 'carefree.' From 'se-,' which means 'apart,' and 'cura,' which means 'care.' Essentially, if one is 'sure,' this means they are 'free from having to take a decision.' 'Uncertain' is therefore the state of an agent who is at a turning point, and is about to make a decision. It is clear then that decision-making and uncertainty are deeply intertwined. 

Ancient Greeks had a name for such a turning point: 'kairos.' 'Kairos' represents the 'fleeting moment,' which must be grasped by the tuft of hair on the personified forehead of the fleeting opportunity, otherwise the moment is gone and cannot be recaptured. In contrast 'chronos' represents the chronological or sequential time. 

Immediately behind 'Kairos' representations typically feature a desperate figure who follows: 'Metanoia,' representing 'afterthought' or, in modern terms, 'buyer's remorse.' "Metanoia exacts punishment for what has and has not been done so that people regrets it." (Ausonius). No matter what one chooses, an alternative always exists, otherwise there would be no choice. It is this (or these) alternative(s) which generates regret. Regret is an intrinsic part of decision making. Because of regret, one may say that - from a present viewpoint - there is not a single future, rather there are multiple possible futures. 

This counterintuitive nature of time has been discussed in a number of popular science books. In decision science, these multiple possible futures are typically represented by means of a tool called 'scenario tree.' A person may use this tool to consider what futures may occur, which are represented by the branches, and the degree of regret faced in each of these possible futures. Decisions would then be taken, which ensure regret remains under control across the scenario tree. These decision points - effectively our kairoi - are represented as round nodes. However, considering the degree of regret faced in each of these possible futures is often not enough. Instead, decision makers often rely on probability to associate to each of these possible futures its likelihood of being realised. 

Probability is a relatively recent development in science. Its origins can be traced to the early Renaissance. However, the foundations of Probability Theory as we know it today were only laid in the mid-1600s. Once more, to fully grasp the meaning of a word, it is helpful to consider its etymology. 'Probable' comes from Latin 'probabilis' which means 'provable.' In a moment, we will see that there are deep connections between the idea of 'provability' and the concept of 'probability.' 

If you are now wondering whether any of this makes any sense in practice, you may want to consider reading this perspective paper published in Science. Uncertainty Modeling is a matter of great concern for policy makers. 

Interestingly, scenario analysis is not limited to the study of the future. Archaeologists use a similar conceptual framework to capture potential interpretations that are compatible with their field findings. This reveals an intrinsic symmetry that uncertainty displays about the present: like the future, the past is also uncertain. 

Scenario trees are useful to capture what Donald Rumsfeld called the "known unknowns:" the things that we don't know and that we know that we don't know. However they become futile when we consider the so-called "unknown unknowns:" the things that we don't know and that we don't know that we don't know. This is the realm of Radical Uncertainty, which is discussed in a recent work by John Kay and Mervin King. Kay and King emphasise that, under radical uncertainty, the use of the narrative and respect for diverse views can generate a better understanding of "what's going on here" than an overalliance on rigid econometric models. Models can still provide a sense of direction and insight, but the sole reliance on them may not allow us to see the wood for the trees. 

But what are these models Kay and King refer to? 'Analytics' is a concept that has been inflated with a plethora of meanings so that it becomes difficult to understand exactly what each of us means when we refer to it. The Cambridge Dictionary defines 'analytics' as "a process in which a computer examines information using mathematical methods in order to find useful patterns." However, this appears to be quite a restrictive definition for our purposes. 

To better understand the nature of Analytics, it is useful to observe that Analytics is often broken down into three parts: descriptive, predictive, and prescriptive. Descriptive Analytics is concerned with answering the question: what happened? Predictive Analytics is concerned with answering the question: what will happen? Prescriptive analytics is concerned with answering the question: how can we make it happen? These are clearly complex questions that cannot be answered by mere number crunching on a computer. To answer these questions, a decision maker must leverage soft as well as hard skills. 

Many tend to think that the 'analytics phenomenon' is a recent development related to widespread availability of computing power. However, in his work 'de Inventio,' the Roman philosopher Cicero states that "there are three parts to Prudence: memory, intelligence, and foresight." It is clear that 'memory' is the skill required to answer the question "what happened?" 'Foresight,' that required to answer the question: "what will happen?" And 'intelligence,' that required to answer the question: "how can we make it happen?" It appears then that Analytics is just a contemporary rebranding of an art that has been known for millennia. Prudentia is the ability to govern and discipline oneself by the use of reason. Inventio is the central canon of Rhetoric, a method devoted to systematic search for arguments. Incidentally, 'inventio' also means 'inventory.' In fact, when a new argument is found, it is 'invented' in the sense of 'added to the inventory of arguments.' 

Prudential and Invention are the foundations upon which the art of Rhetoric stands. Ciceronian inventio intended as a systematic search for arguments is likely to have influenced the work of Ramon Lull. Lull invented a philosophical system known as 'the Art.' The Art was conceived as a type of universal logic to prove the truth of Christian doctrine to interlocutors of all faiths and nationalities. It consists of a set of general principles, and combinatorial operations, and it is illustrated with diagrams. For this reason, Lull is considered the father of Automated Reasoning, a part of Artificial Intelligence that is indeed concerned with systematic search for arguments. 

Automated Reasoning and Machine Learning are the two key branches of Artificial Intelligence. Machine Learning is inductive and empirical in nature. It starts from a data set, and it learns to perform a task such as recognizing a picture, or reading a piece of text. 

Automated Reasoning is deductive and theoretical. It starts from a model of the world, and it deduces its properties by logical reasoning. This is achieved by leveraging mathematical models that are processed by using techniques such as mathematical programming and constraint programming. 

In the early 2000s Stochastic Constraint Programming was introduced as a framework to carry out automated reasoning under uncertainty. In 2012, my co-authors and I extended Stochastic Constraint Programming by introducing Global Chance-constraints. A framework that makes it possible to apply existing constraint programming algorithms to general problems of decision making under uncertainty. In a nutshell, our approach explores provability across scenarios, and builds up a solution that is 'provable' - which means feasible - across a sufficient number of scenarios. As anticipated, there is a close connection between 'provability' and 'probability' in the context of decision making under uncertainty. 

In 2015, we also introduced Confidence-based Reasoning. An approximation framework to find solutions that satisfy constraints with confidence probability alpha, and within a margin of error theta. This represents one of our contributions to Inventio intended as a systematic search for arguments. 

'Inventio' however also means 'inventory' - a domain of research to which we have also extensively contributed. Inventory management finds its roots into the practice of late medieval and early Renaissance merchants. The invention of double-entry bookkeeping is typically attributed to Luca Pacioli. However, what Pacioli did was to exploit Gutenberg's technology to disseminate practices that had been in use among venetian merchants for a long time. In his Tractatus, he states "in order to conduct a business properly a person must possess sufficient capital or credit, be a good accountant and bookkeeper, and possess a proper bookkeeping system." The treatise is divided into two main sections: 'the inventory' and 'the disposition.' The influence of Cicero is apparent. In 'the inventory' Pacioli describes the process by which a business physically reviews its entire inventory. This practice is known today as the 'physical inventory.' In 'the disposition' he describes the necessary books and rules to implement double-entry bookkeeping. 

This work represents a quantum leap in the realm of descriptive inventory analytics. However, no progress was made in the realm of predictive and prescriptive inventory analytics until late 1800. The economic order quantity problem introduced by Ford Harris in 1913, and the dynamic version of the economic lot size model discussed by Wagner and Whitin in 1958 represent two milestones in the realm of deterministic prescriptive inventory analytics. Surprisingly, the first prescriptive inventory analytics model ever discussed operates under uncertainty. This is the so-called Newsvendor Problem introduced by Edgeworth in his 'Mathematical Theory of Banking' published in 1888. The problem was inspired by the challenge faced by a bank that needs to decide how much cash to stock on a given day to meet random withdrawals from customers. The extensive literature on the Newsvendor Problem flows directly from Edgewood's formulation, and so does the one on quantile regression. It was not until the 50s that dynamic inventory systems subject to random demand started to be investigated by Arrow, Harris and Marshak. In 1960, a fundamental result was published by Scarf: the optimality of (s,S) policies for the dynamic inventory problem. In this policy, one orders up to capital S when inventory level falls below small s, and do not order otherwise. 

Nowadays (s,S) policies feature prominently in all contemporary enterprise resource planning systems; but how do we compute the order up to level capital S and the reorder point small s? Between the 80s and the 90s computational approaches were proposed which achieved optimality gaps of around four to five percent. In 2014, building upon research spanning over two decades, we published a work outlining a general purpose approach for piecewise linearizing the so-called first order loss functions - a key building block of any stochastic inventory control model. By leveraging these results, in 2018 we introduced the first mixed integer linear programming approach for computing (s,S) policy parameters. Our approach achieves optimality gaps that are generally below 0.3 percent; it hence represents the current state of the art for computing (s,S)) policies. 

The (s,S) policy - albeit cost optimal - is only one of the possible policies that can be operationalized to control stochastic inventory systems. In their 1988 work Bookbinder and Tan discussed the so-called Static-Dynamic Uncertainty control strategy, which represents a possible alternative. The Static-Dynamic Uncertainty control strategy takes two possible forms. In the (R,S) policy replenishment timings are fixed statically at the beginning of the planning horizon, while replenishment quantities are decided dynamically just before replenishment order is issued, so to raise the inventory level up to capital S. In the (s,Q) policy order quantities are fixed statically at the beginning of the planning Horizon, while replenishment timings are decided dynamically: an order is issued whenever the inventory level falls below small s. Both these policies are important because they can help reducing the so-called nervousness of the control action, that is the unpredictability of replenishment order timing and/or sizes. In our 2015 work, by leveraging once more our piecewise linearization strategy, we endeavour to tackle the challenge of computing near optimal (R,S) policy parameters. More recently, in 2022 we developed a cognate approach for computing near optimal (s,Q) policy parameters. All previous work assume that random demand is independently distributed across periods. However, in reality demand often displays correlation across periods. The presence of correlation makes the problem of determining good control strategies extremely challenging. By leveraging properties of the variance of the multi-normal distribution, along with our piecewise linearization approach, we were able to develop an efficient and effective mathematical programming-based solution method for stochastic inventory control under correlated demand. Once more our approach features optimality gaps that generally remain below one percent. A compendium of results from inventory control theory which includes these recent developments is presented in my book inventory analytics published in 2021. 

We have analyzed the nature of uncertainty in its close relationship to decision making. We have then moved our attention to analytics and investigated its origins, which surprisingly date back to Rhetoric, the Roman art of persuasion. We have seen how Rhetoric stands on two pillars Prudentia and Inventio; and that Inventio conceals a productive ambiguity that links its meaning to Automated Reasoning - intended as a systematic search for arguments -  as well as to Inventory Management. This allowed me to showcase my contributions to both these fields. Of course, none of these contributions would have been possible without the help of my international collaborators and of my PhD students; all of whom I wholeheartedly thank.