An analytical method often considers the optimization of a precise objective variable (e.g. production cost, profit, etc.), while the simulation model limits itself to evaluating, moment by moment, the value of each variable present in the model based on its causal relationships (deterministic or stochastic) with the other variables. It follows that the choice of the objective function of an analytical method reflects the degree of specificity that you want to give to the problem through the variables used to determine the final result to be optimized. For example, in maximizing the profit of a given problem, the calculation function of that indicator could be trivially modeled as:

Profit = (price – variable cost) * quantity – fixed costs

or it may need to be furtherly sophisticated, detailing the macro variables in their primary (e.g. costs of production factors) or secondary components. The choice of the correct level of detail, linked to the “free” variables of the problem, therefore influences the system of constraints and the number of calculations (derivatives of the objective function) that must be considered.
This aspect formally constitutes an a priori choice when selecting an analytical model: the incorrect structuring of the objective function can lead to partially or completely review the system of constraints and variables considered by the problem. Furthermore, the level of detail considered in the objective function is however limited by the very nature of the algorithm: the calculation time and the goodness/applicability of the theoretical result are heavily conditioned by the number of variables and analysis dimensions included in the algorithm. This can lead some applications to propose “degenerate” solutions, that is, valid from an algorithmic point of view but difficult to implement in the reality considered.

On the contrary, a simulation model allows you to generate results based on a macro structure of links that regulate how the model variables interact with each other. This allows you to develop the first simplified “drafts” of the model and quickly test them to verify their effectiveness and the reliability of the results obtained. Based on the analysis of these results, it is then possible to proceed to define further variables to add to the model to increase the capacity and correctness of the estimate.
This development mode by addition often allows for significant time savings and satisfactory forecasting results, effectively enabling agile model development.

The number of variables in play is clearly the factor with the main impact on the complexity of the problem. Often this aspect is not known a priori, as the variables emerge gradually as the problem is analyzed and modelled.
From this point of view, the variables considered by an analytical method can be the same as those of a simulation model and vice versa. The difference between the two approaches in this case mainly concerns the clarification of how these variables interact: in the simulation model a reasoning perspective for classes of variables can be adopted while in the analytical method it is necessary to define and verify new constraints for each type of added variable.

A second aspect that has a strong impact on the complexity of the analyzed problem concerns the so-called analysis dimensions. By dimensions or classes we mean the number of sets of values that represent the different types of variables considered. Typical classes that we can find in any model (analytical and simulation) inherent to the supply chain are: products, nodes, resources, paths, time, etc. In other words, classes represent the different indexes that we can find within the objective function and constraints of an analytical algorithm.
From this point of view, the complexity of the problem grows as the classes considered increase, and there is no difference at this level between the methods.
However, although usually the number of dimensions considered for solving a problem is equivalent, the simulation differs heavily from the analytical method due to the possibility of considering time as a continuous and not discrete dimension of analysis. While each class is usually characterized by a finite (discrete) set of elements, time is an exception as it is a continuous dimension: the possibility of considering it in these terms entails various benefits for the decision maker, who not only obtains a result but is able to verify the evolution of a variable over time and analyze the main events that have modified it.
Obviously this peculiarity can also be partially captured in an analytical model by carrying out the so-called bucketization of time (division of the time horizon into equal instants) with a very small level of detail (e.g. seconds). However, this action strongly affects resolution times and is not easily usable in practice.

The complexity of a system is measurable by identifying its constituent variables, as previously described, and defining their behavior. Often the behavior of the variables representing the observed phenomena is hypothesized for convenience and simplicity of resolution, effectively compromising the goodness and adaptability of the solution to the context studied.
To give an example, often the demand recorded at a supply chain node at a certain moment is seen as an average value and is assumed to be constant over the period considered. This may prove to be an acceptable simplification for certain situations but in other cases it does not allow us to grasp any critical issues inherent to the proposed solution.  Hypothesizing the probability distribution of any event already constitutes a simplification of the nature of the problem studied. Through simulation it is possible to consider continuous stochastic variables capable of approximating the analyzed reality as faithfully as possible. The importance of developing reasoning and solutions on a stochastic basis is fundamental, as:

• It allows you to analyze the results by case study (e.g. best case, worst case, most probable case), broadening the vision of decision makers and the range of actions that can be implemented (think for example of different risk management plans). This is made possible thanks to the possibility of repeating the simulation a large number of times, always generating different results from which conclusions can then be deduced according to statistical inference processes.
• It allows you to associate certain results and factors with probabilities of occurrence, which help to analyze the problem and highlight particular critical issues that must be specifically addressed.
• It allows prioritizing management efforts and resources according to a quantitative criterion.
• It allows you to link variables controllable by the decision maker (endogenous) with variables exogenous to the system, defining its degree of action on the final result.
• It teaches us to accept deviations in results from targets as part of reality and not as a result of a lack of effectiveness or efficiency in the work performed.

The optimality of a solution indicates its property of being better than other solutions according to a given criterion. It follows that optimality is a characteristic that can be attributed to a solution based on a series of criteria defined by the decision maker.
As already mentioned, simulation is not used to determine optimal solutions but rather results intended as a consequence of the behavior of the variables considered. In practice, however, it may happen that some parts of the simulation model are subject to optimization: it is in fact possible to execute particular optimization algorithms capable of to select the best result for a given parameter of the problem, under the appropriate conditions. In these applications, rather than the optimal solution per se, the decision maker is often interested in evaluating the change in the optimal solution as the model conditions vary.

Consider, for example, the determination of safety stocks within a logistics network. Theory teaches us that the calculation of the amount of such stocks at a given instant can be obtained by deriving a cost function which takes into account a series of costs linked to the management of stocks and the failure to provide a service (relating to the lack of stock ). This allows you to identify a unique value, given a series of values set for the input variables (such as demand or desired service level).
In a similar application, but developed with a simulation model, the decision maker has at his disposal a series of levers on which he can act (e.g. service level, storage capacity, financial immobilization) and a series of exogenous variables on which he cannot act (e.g. variability in demand). He will therefore be able to simulate different plausible scenarios to check what happens to different indicators, varying from time to time the sets of values that can be used to define the available levers .

A factor that can be objectively measured is the precision of the estimate made by the model. In this case, as happens with many statistical models (e.g. regressive, correlative, etc.), precision can be measured by the deviation recorded between the expected result (obtained through the model) and the actual result. In statistics, the sum of the squares of these deviations (MSE method) therefore constitutes the term to be optimized (minimized) to refine the model’s estimation capacity by acting on its parameters. In simulation, however, this procedure is not easily implementable because there are often many links between different “intermediate” variables that contribute to determining the result of the “final” variables (results of the simulation model). These intermediate variables cannot be directly verified as the final data to make a comparison is often missing.

Let’s take the example of safety stocks in a logistics network again: let’s say that the simulation model allows an estimate of the service level, based on a history of orders, of 85% while in reality a service level was measured by 90%. It is clear that the result of the model can be traced back to a series of different factors, such as: network reactivity (order processing time), transportation times, initial stock level, stock management policies, etc. These variables, which are placed in an “intermediate” layer between the primary variable of the demand and between the result of the model relating to the service, are often difficult to verify with final data. So how can the simulation model guarantee the accuracy of its results? In the face of deviations, the model allows us to trace the various component variables of the result and verify their status (i.e. the value assumed at each instant of time) to highlight anomalous situations capable of explaining the divergence of the result obtained compared to the expected according to a cause-effect logic. By doing so it is possible to correct the simulation model in progress, until obtaining a tool capable of faithfully replicating reality. If at the end of the corrections, which are often necessary to capture specific aspects of the company’s business, the model still presents significant variances in its results, it is a good idea to check under which hypotheses and reasoning the final results were calculated. In fact, it is not uncommon that in reality we often tend to make calculations and simplifications on the starting data (e.g. to remove a series of orders when calculating the service level for a specific reason), thus obtaining results that may differ from the ones of the model which however are more realistic in practice.