## 3 Materials and Methods
### Proposed Approach
#### 3.1.1 Prescriptive Module  
A parametrized DLM based on mixed-integer linear programming (MILP) is proposed to build the prescriptive module. The model is constructed based on several key assumptions, outlined as follows:  
* The demand for the components allocated among suppliers is predetermined and constant for the designated future period.
* The unit purchasing cost varies among suppliers, and the total purchasing cost escalates linearly with the quantity ordered without any quantity discounts.
* The unit costs associated with untimely deliveries differ depending on whether the delivery is early or late. These costs are contingent on the components and remain consistent across suppliers. Additionally, the overall cost attributed to untimely deliveries increases linearly with the quantity delivered late and the duration of the delay.
* There is no relationship between the ordered quantity and the future value of the supplier delivery performance until orders remain within the supplier capacity limits.  
Following these assumptions, the principal sets, parameters, decision variables, constraints, and components constituting the objective function to be minimized are enumerated below:  
* \(\text{I}=\{1,\ldots,\text{i},\ldots,\text{M}\}\): set of suppliers
* \(\text{P}=\{1,\ldots,\text{t},\ldots,\text{T}\}\): set of time period  
Parameters  
* \(C_{i}^{period}\): maximum period capacity of supplier \(i\)
* \(c_{i}^{order\ emission}\): unitary order emission cost of supplier \(i\)
* \(c_{i}^{purchasing}\): unitary purchasing cost of supplier \(i\)
* \(c^{delay}\): unitary cost related to late delivery of a component
* \(c^{dvance}\): unitary cost related to advance delivery of a component
* \(d_{t}\): demand of the component that needs to be ordered for day \(t\)
* \(f_{it}^{delay}\): number of days of delays predicted based on the predictive module for the number of components ordered from supplier \(i\) for day \(t\)
* \(f_{it}^{dvance}\): number of days of advance delivery predicted by the predictive module for the number of components ordered from supplier \(i\) for day \(t\)
* \(\theta_{i}^{data\ benefiti}\): tunable parameters expressing the advantages of sending orders to supplier \(i\) to acquire new data. This parameter models the data acquisition benefit for supplier \(i\)\(M\): positive large number  
Decision variables  
\(Y_{it}\): amount of quantity ordered from supplier \(i\) for day \(t\)  
\(Z_{it}\): binary variable equal to 1 if an order is sent to supplier \(i\) for day \(t\), 0 otherwise  
Objective function  
\[\text{M}in\sum_{t=0}^{T}\sum_{i=1}^{M}Y_{it}\left(c_{i}^{\text{ }parchasing}+c^{\text{ }delay}f_{it}^{\text{ }delay}+c^{\text{ }distance}f_{it}^{\text{ }distance}\right)+Z_{it}\left(c_{i}^{\text{ }order\ emission}-g_{i}^{\text{ }data\ benefit}\right) \tag{1}\]  
In Equation (1), the objective function is aimed at finding the optimal trade-off between purchasing costs and costs related to non-punctual delivery by suppliers, with the order emission cost on one side and the data acquisition benefits on the other.  
Constraints  
\[\sum_{t=0}^{T}Y_{it}\leq C_{i}^{\text{period}}\ \forall i \tag{2}\]  
\[\sum_{i=1}^{M}Y_{it}=d_{t}\ \forall t \tag{3}\]  
\[Y_{it}\leq M*Z_{it}\ \forall i,\forall t \tag{4}\]  
\[Y_{it}\geq 0,\ \forall i,\forall t \tag{5}\]  
\[Z_{it}\ Binary\ \forall i,\forall t \tag{6}\]  
Equation (2) is devised to ensure that the maximum quantity ordered from each supplier remains within acceptable limits. Equation (3) guarantees fulfilment of the daily demand for the component. Equation (4) ensures that a certain quantity of a specific component can be acquired from supplier \(i\) in period \(t\) only if an order is sent to supplier \(i\) in period \(t\). Lastly, Equations (5) and (6) mandate, respectively, that the orders dispatched to each supplier during each period must exceed zero, and that the variable \(Z_{it}\) has to be binary.