## 2 Background Literature
### Employing Shannon's Entropy as a Proxy to Complexity in SCs  
Shannon entropy can be helpful for evaluating complexity by measuring the uncertainty or average unexpectedness associated with a random variable or, in the context of information theory, the amount of information contained in a message [50]. The flow of information exchanged between agents within an SC can be considered as a message, in which the necessity of the next symbol or lack thereof (a piece of managerial information) is the outcome of a random variable. Equation (1) gives [PERSON]'s entropy \(H(P)\), subject to Equation (2), for the discrete outcomes of a random variable that will be employed in this study. The resulting unit is expressed in bits [7]. \[H(P)=-\sum_{i=1}^{n}p_{i}\cdot log(p_{i}) \tag{1}\]  
\[\sum_{i=1}^{n}p_{i}=1 \tag{2}\]  
where:  
\(H(P)\) = entropy of a random variable;  
\(p_{i}\) is the probability of the \(i\)th outcome;  
\(n\) is the number of possible outcomes. The function \(H(P)\) serves as a measure to quantify the level of uncertainty inherent in a probability distribution. This study handles the amount of information necessary to coordinate the actions necessary to meet orders' requirements within SCs. Higher values of \(H(P)\) indicate increased uncertainty or randomness in the distribution, while lower values suggest a higher degree of predictability and reduced uncertainty. Thispredictability reduces the need for coordination, leading to decreased internal complexity. In simpler terms, a higher \(H(P)\) means more information is required to dissipate uncertainty in fulfilling orders, whereas a lower \(H(P)\) implies the opposite. When dealing with systems exceeding a single dimension [51], such as the multiple agents actuating in an SC, Shannon's entropy adopts the multidimensional form presented in Equation (3). \[H(P)=-\sum_{k=1}^{m}\ldots\sum_{i=1}^{n}p_{i..k}\cdot log(p_{i..k}) \tag{3}\]  
where:  
\(n\) is the number of possible outcomes for each variable;  
\(m\) is the number of elements in a given dimension of the problem;  
... are further dimensions of the problem;  
\(p_{i..k}\) are the probabilities of the possible outcomes for the variables. For a continuous random variable \(X\) with a continuous probability distribution \(p(x)\), Equation (4) shows the Shannon entropy, subject to Equation (5) [52]. \[H(X)=-\int_{-\infty}^{\infty}p(x)\cdot logp(x)dx \tag{4}\]  
\[\int_{-\infty}^{\infty}p(x)=1 \tag{5}\]  
where:  
\(x\) is the continuous random variable whose entropy is being calculated; and  
\(p(x)\) is the probability density function of \(X\). A case of interest for this study occurs when \(n=2\), which is the so-called binary Shannon entropy, when the maximum entropy occurs when \(p_{i}=0.5\). In this case, the Shannon entropy assumes the form shown in Equation (6). Figure 2 showcases the binary \(H(P)\), which is a concave function with a maximum at \(p=0.5\) and an outcome of \(H_{bin}(p)=1\) bit [7]. \[H_{bin}(p)=-p\cdot log(p)-(1-p)\cdot log(1-p) \tag{6}\]  
Entropy is inversely proportional to the diversity of possible outcomes. As the likelihood of a specific group of symbols occurring becomes dominant, the overall entropy of the system decreases [7]. After applying Shannon entropy to a random variable, it is useful to compare the result to the maximum possible entropy, which occurs when the probabilities are uniformly distributed. Taking the derivative of Equation (1) and setting it equal to zero yields the point of maximum entropy. Treating Shannon's entropy as a relative or  
Figure 2: The binary Shannon entropy function \(H_{bin}(p)\). Source: Authors’ own creation based on [7]. percentage rather than an absolute measurement is beneficial, as entropy is influenced not only by the probability distribution but also by the number of possible outcomes and symbols in the message. A system with numerous outcomes and symbols should exhibit a higher absolute value for entropy.