Table 3.

Notations of the proposed multi-period PISP model.

Notation	Description
Sets & Indices
$m,m^{\prime }\in M$	Set of nodes, including manufacturers and hospitals in all regions
$K$ & $H$	Set of manufacturers and hospitals in all regions
$K_i\left(m\right)$	Subsets of manufacturers in region $i$; $K=\bigcup _i{K}_i\left(m\right)$
$H_i\left(m\right)$	Subsets of hospitals in region $i$; $H=\bigcup _i{H}_i\left(m\right)$
$j\in J$	set of suppliers of raw materials
$p\in P$	Set of medical products, including two sets of consumable $P^{\prime }$ (hydro-alcoholic gel, mask, medical clothes, and ordinary beds) and reusable $P^{″}$ (ICU beds and ventilator)
$t\in T$	Set of time periods
Parameters
$L$	Lead-time periods as the occupancy time, at which the reusable products are reconsidered as available inventory
${\gamma }_p$	Conversion rate of raw materials to manufacture product $p$
$D_{pmt}$	Demand for product $p$ in hospital $m$ at period $t$
$U_{pj}$	Capacity limitation for product $p$ at supplier $j$
$Q_{pm}$	Capacity limitation for product $p$ at manufacturer $m$
$O{C}_j$	Ordering cost of product $p$ at supplier $j$
$S{C}_{pm}$	Production cost of product $p$ at manufacturer $m$
$P{C}_{pm}$	Unit production cost of product $p$ at manufacturer $m$
$T{C}_{pjm}$	Transportation cost per unit of product $p$ from supplier $j$ to manufacturer $m$
$CU$	Unit cost of unmet demand at hospitals
$I_{pm}^0$	Initial inventory of product $p$ at manufacturer/hospital $m$ at the beginning of the planning horizon
$I{C}_{pm}$	Unit inventory holding cost of product $p$ at manufacturer/hospital $m$
$I_{pm}^{\max }$	Maximum inventory capacity for product $p$ at manufacturer/hospital $m$
$G_{pm}^{\max }$	Maximum sharing capacity for product $p$ at hospital $m$
$T{Q}_p$	Total sharing capacity for product $p$ over all regions
${\mu }_p$	Allowed demand surplus (%) for product $p$ at hospitals that can be met through the sharing system
$T{C}_{pm{m}^{\prime }}$	Unit transportation cost of product $p$ shared among hospitals $m$ and $m^{\prime }$ during the sharing process
Decision variables
$w_{jt}$	1 if an order is placed to supplier $j$ at period $t$; 0 otherwise
$y_{pmt}$	1 if product $p$ is set up for production in manufacturer $m$ at period $t$
$b_{pmt}$	Net service balance of product $p$ in hospital $m$ at the beginning of period $t$ before transferring the products in the sharing process; $b_{pmt}=b_{pmt}^{+}+b_{pmt}^{-}$, where $b_{pmt}^{+}$ and $b_{pmt}^{-}$ are positive and negative parts of net service balance $b_{pmt}$ (i.e., $b_{pmt}^{+}=\max \{0,b_{pmt}\}$ and $b_{pmt}^{-}$ = $\min \{0,b_{pmt}\}$)
$v_m$	1 if $b_{pmt}>0$; 0 otherwise
$x_{pmt}$	Production quantity of product $p$ in manufacturer $m$ in period $t$
$u_{pjmt}$	Quantity of raw material for product $p$ transferred from supplier $j$ to manufacturer $m$ in period $t$
$q_{m{m}^{\prime }}^{pt}$	Quantity of product $p$ transferred from manufacturer $m$ to hospital $m^{\prime }$ at the beginning of period $t$
$I_{pmt}$	Inventory of product $p$ in node $m$ at the end of period $t$
$g_{pmt}$	Available quantity of product $p$ to be supplied by hospital $m$ in period $t$ to satisfy its internal demand and external sharing
$s_{m{m}^{\prime }}^{pt}$	Quantity of product $p$ shared from hospital $m$ to hospital $m^{\prime }$ in period $t$; $s_{mm}^{pt}=0$ and $s_{m{m}^{\prime }}^{pt}=-s_{m^{\prime }m}^{pt}$, where ${\left(s_{m{m}^{\prime }}^{pt}\right)}^{+}$ and ${\left(s_{m{m}^{\prime }}^{pt}\right)}^{-}$ are the positive and negative parts of shared products (i.e., ${\left(s_{m{m}^{\prime }}^{pt}\right)}^{+}=\max \{0,s_{m{m}^{\prime }}^{pt}\}$ and ${\left(s_{m{m}^{\prime }}^{pt}\right)}^{-}=\min \{0,s_{m{m}^{\prime }}^{pt}\}$ for $m\ne m^{\prime }$)
$S_{pmt}$	Amount of unmet demand of product $p$ in hospital $m$ at the end of period $t$
$S_{\max }$	Maximum unmet demand among all hospitals