## Appendix A Proof  
Proof of Proposition 1.: For a demand location, the inventory cost before the replenishment arrival is not affected by the replenishment quantity, it only depends on the initial inventory level. So the sub-problem is determining replenishment quantities to minimize the inventory cost after the replenishment arrival.  
If the demand location is replenished, there are two situations:  
1. The demand location is replenished by one replenishment mode. If the initial inventory is enough, where \[h(I_{0}-H\bar{d}+H\bar{d})\geq-s(I_{0}-H\bar{d}-H\bar{d}).\] (11) Let \(\lambda=(s+h)\bar{d}+(s-h)\bar{d}\), the optimal replenishment quantity is 0 when \[I_{0}\geq\frac{\lambda}{s+h}H.\] (12) If \(I_{0}<\lambda H/(s+h)\) there is a turning point \(t_{1}\) between the replenishment arrival time and the inventory planning horizon \(H\). Where the inventory cost is from the holding cost turning to the shortage cost under the worst case. The turning point \(t_{1}\) and the replenishment quantity \(q\) satisfy: \[h(I_{0}+q-t_{1}\bar{d}+t_{1}\bar{d})=-s(I_{0}+q-t_{1}\bar{d}-t_{1}\bar{d}).\] (13) Then, \[t_{1}=\frac{(s+h)(I_{0}+q)}{(s+h)\bar{d}+(s-h)\bar{d}}.\] (14) For any replenishment lead time \(l\), the inventory cost after the replenishment is \[IC^{replenished}=\int_{l}^{t_{1}}h(I_{0}+q-t\bar{d}+t\bar{d})dt-\int_{t_{1}}^{ H}s(I_{0}+q-t\bar{d}-t\bar{d})dt.\] (15) The optimal replenishment quantity is \[q=\arg\Bigg{\{}\frac{dIC^{replenished}}{dq}=0\Bigg{\}}.\] (16) The closed-form solution for optimal replenishment quantity is \[q=\Bigg{\{}\frac{(sH+hl)}{(s+h)^{2}}\lambda-I_{i0}\qquad I_{i0}<\frac{\lambda }{s+h}H\] (17) 2. The demand location is replenished by both two replenishment modes. If the initial inventory is enough before the slower replenishment arrival, where \[h(I_{0}-l^{s}\bar{d}+l^{s}\bar{d})\geq-s(I_{0}-l^{s}\bar{d}-l^{s}\bar{d}).\] (18) The optimal faster replenishment quantity is 0 when \[I_{0}\geq\frac{\lambda}{s+h}l^{s}.\] (19) Similarly, the optimal slower replenishment quantity is 0 when \[I_{0}+q^{\prime}\geq\frac{\lambda}{s+h}H.\] (20)For the faster arrival replenishment, if \(I_{0}<\lambda l^{s}/(s+h)\), there is a turning point \(t_{1}\) between the twice replenishment.  
\[t_{1}=\frac{(s+h)(I_{0}+q^{f})}{(s+h)\bar{d}+(s-h)\bar{d}}. \tag{100}\]  
For any replenishment lead time \(l^{f}\), the inventory cost between the faster and the slower replenishment arrival time is  
\[IC^{replenished_{1}}=\int_{ll}^{t_{1}}h(I_{0}+q^{f}-t\bar{d}+t\bar{d})dt-\int_{ t_{1}}^{s}s(I_{0}+q^{f}-t\bar{d}-t\bar{d})dt. \tag{101}\]  
The closed-form solution for the faster replenishment quantity is  
\[q_{i}^{f}=\begin{cases}\frac{(s!s+hl^{f})}{(s+h)^{2}}\lambda-I_{i0}&I_{i0}< \frac{\lambda}{s+h}l^{s}\\ 0&I_{i0}\geq\frac{\lambda}{s+h}l^{s}\end{cases} \tag{102}\]  
For the slower arrival replenishment, if \(I_{0}+q^{f}<\lambda H/(s+h)\), there is a turning point \(t_{2}\) between the slower replenishment arrival time and the inventory planning horizon \(H\).  
\[t_{2}=\frac{(s+h)(I_{0}+q^{f}+q^{s})}{(s+h)\bar{d}+(s-h)\bar{d}}. \tag{103}\]  
For any replenishment lead time \(l^{s}\), the inventory cost after the slower replenishment arrival is  
\[IC^{replenished_{2}}=\int_{l^{s}}^{t_{2}}h(I_{0}+q^{f}+q^{s}-t\bar{d}+t\bar{d} )dt-\int_{t_{2}}^{H}s(I_{0}+q^{f}+q^{s}-t\bar{d}-t\bar{d})dt. \tag{104}\]  
The closed-form solution for the slower replenishment quantity is  
\[q_{i}^{s}=\begin{cases}\frac{(sH+hl^{s})}{(s+h)^{2}}\lambda-I_{i0}-q_{i}^{f}&I _{i0}+q_{i}^{f}<\frac{\lambda}{s+h}H\\ 0&I_{i0}+q_{i}^{f}\geq\frac{\lambda}{s+h}H\end{cases} \tag{105}\]