## 3 Materials and Methods
### Evaluation Metrics  
Four economic metrics are considered to compare the cost when implementing order allocation decisions according to the proposed approach with those resulting from the benchmark approaches:  
\[ \begin{split}\underset{\Delta(proposed,benchmark)}{ \overset{PUTRCHASING\ COST}{\Delta(proposed,benchmark)}}&=\frac{ \sum_{t^{\prime}=0}^{T}\sum_{t^{\prime}\geq t>t^{\prime}}^{T}\sum_{i=1}^{M} \Big{(}\dot{Y}_{itt^{\prime}}^{proposed}-\dot{Y}_{itt^{\prime}}^{benchmark}\Big{)}c^{ purchasing}}{\sum_{t^{\prime}=0}^{T}\sum_{t^{\prime}\geq t>t^{\prime}}^{T}\sum_{i=1}^{M} \dot{Y}_{itt^{\prime}}^{benchmark}c^{parchasing}}\end{split}  \tag{7}\]  
\[ \begin{split}\underset{\Delta(proposed,benchmark)}{ \overset{DELIVERY\ COST}{\Delta(proposed,benchmark)}}&=\frac{\sum_{t^{ \prime}=0}^{T}\sum_{t^{\prime}\geq t>t^{\prime}}^{T}\sum_{i=1}^{M}\Big{(} \dot{Y}_{itt^{\prime}}^{proposed}-\dot{Y}_{itt^{\prime}}^{benchmark}\Big{)}\Big{(}c^{delay} true_{it}^{delay}+c^{advance}true_{it}^{endvance}\Big{)}}{\sum_{t^{\prime}=0}^{T}\sum_{t^{\prime}\geq t >t^{\prime}}^{T}\sum_{i=1}^{M}\dot{Y}_{itt^{\prime}}^{benchmark}\Big{(}c^{delay}true_{it}^{delay}+c^{advance}true_{it}^{ endvance}\Big{)}}\end{split}  \tag{8}\]  
\[ \begin{split}\underset{\Delta(proposed,benchmark)}{ \overset{DELIVERY\ COST}{\Delta(proposed,benchmark)}}&=\frac{\sum_{t^{ \prime}=0}^{T}\sum_{t^{\prime}\geq t>t^{\prime}}^{T}\sum_{i=1}^{M}\Big{(} \dot{Y}_{itt^{\prime}}^{proposed}-\dot{Y}_{itt^{\prime}}^{benchmark}\Big{)}\Big{(}c^{delay}true_{it}^{delay}+c^{advance}true_{it}^{ endvance}\Big{)}}{\sum_{t^{\prime}=0}^{T}\sum_{t^{\prime}\geq t >t^{\prime}}^{T}\sum_{i=1}^{M}\dot{Y}_{itt^{\prime}}^{benchmark}\Big{(}c^{delay}true_{it}^{delay}+c^{advance}true_{it}^{ endvance}\Big{)}}\end{split}  \tag{9}\]  
\[ \begin{split}\underset{\Delta(proposed,benchmark)}{ \overset{OPERMISO\ COST}{\Delta(proposed,benchmark)}}&=\frac{\sum_{t^{ \prime}=0}^{T}\sum_{t^{\prime}\geq t>t^{\prime}}^{T}\sum_{i=1}^{M}\Big{(} \dot{Y}_{itt^{\prime}}^{proposed}-\dot{Y}_{itt^{\prime}}^{benchmark}\Big{)}\Big{(}c^{parchasing}+c^{delay}true_{it}^{delay}+c^{advance}true_{it}^{ endvance}\Big{)}}{\sum_{t^{\prime}=0}^{T}\sum_{t^{\prime}\geq t >t^{\prime}}^{T}\sum_{i=1}^{M}\dot{Y}_{itt^{\prime}}^{benchmark}\Big{(}c^{parchasing}+c^{delay}true_{it}^{delay}+c^{advance}true_{it}^{ endvance}\Big{)}+\\ &\frac{\sum_{t^{\prime}=0}^{T}\sum_{t^{\prime}\geq t>t^{\prime}}^{T} \sum_{i=1}^{M}\dot{Y}_{t^{\prime}}^{benchmark}\Big{(}c^{parchasing}+c^{delay}true_{it}^{delay}+c^{advance}true_{it}^{ endvance}\Big{)}+\dot{Z}_{itt^{\prime}}^{benchmark}c^{parchasing}}{\sum_{t^{\prime}=0}^{T}\sum_{t^{\prime}\geq t >t^{\prime}}^{T}\sum_{i=1}^{M}\dot{Y}_{t^{\prime}}^{benchmark}\Big{(}c^{parchasing}+c^{delay}true_{it}^{delay}+c^{advance}true_{it}^{ endvance}\Big{)}+\dot{Z}_{itt^{\prime}}^{benchmark}c^{parchasing}}\end{split}  \tag{10}\]Here, \(Y^{proposed}_{itt^{\prime}}\) represents the optimal quantity to allocate to a specific supplier \(i\) for a specific period \(t\) based on the forecast updated at day \(t^{\prime}\) according to the proposed approach, while \(Z^{proposed}_{itt^{\prime}}\) expresses whether an order is sent to supplier \(i\) in period \(t\) based on forecasts updated at day \(t^{\prime}\) by the proposed approach. Contrarywise, \(Y^{benchmark}_{itt^{\prime}}\) and \(Z^{benchmark}_{itt^{\prime}}\) represent the same optimal quantity identified when considering one of the benchmark approaches reported in Section Benchmarks, respectively. The parameters \(true^{delay}_{it}\) and \(true^{advance}_{it}\) represent the true delivery punctuality reported by supplier \(i\) in period \(t\). These values can be higher or lower than the predicted values reported in the parameters \(f^{delay}_{it}\) and \(f^{advance}_{it}\).