Table 1 Table of notations Sets V Set of...

Table 1

Table of notations

Sets
$V$	Set of indexes for warehouses $v \in V$
$T$	Set of indexes for periods $t \in T$
$I$	Set of indexes for demand points $i \in I$
$F$	Set of indexes for products $f \in F$
Parameters
$λ_{v t}$	The arrival rate of vehicles at the warehouse $v$ in the period $t$
$µ_{v t}$	Unloading rate at the warehouse $v$ at period $t$
$ρ_{v t}$	The traffic intensity of queue formed warehouse $v$ in the period $t$
$k_{v}$	Waiting rooms space for the warehouse $v$
$S_{v}$	Maximum number of servers at the warehouse $v$
${cap}_{v f}$	The capacity of arrival vehicles to the warehouse $v$ for product type f
$p_{i t f}$	The demand of demand point $i$ in the period t for product type f
$Q_{v f}$	The capacity of distribution vehicles of the warehouse $v$ for product type f
$τ_{i l}$	The traveling time between demand point $i$ and demand point $l$
$τ_{i v}^{'}$	The traveling time between demand point $i$ and warehouse $v$
$c_{i l}$	Traveling cost between demand point $i$ and $l$
$w t_{i}$	Importance of demand point $i$
$B_{i}$	Unloading time in the demand point $i$
$ω_{v f}$	Inventory cost of warehouse $v$ for product type $f$
Decision variables
$s_{v}$	Number of allocated servers to warehouse $v$
$w_{v t}$	Average waiting time for vehicles in the warehouse $v$ in the period $t$
$w q_{v t}$	Average waiting time for vehicles in the queues in the warehouse $v$ in the period $t$
$L_{v t}$	The average number of vehicles in the warehouse $v$ in the period $t$
$L q_{v t}$	The average number of vehicles in the queues in the warehouse $v$ in the period $t$
$π_{n}^{v t}$	Probability of existence of n vehicles in the warehouse v in the period t
$I_{v t f}$	Inventory level of product f in the warehouse v at the end of the period t
$z_{i v t}$	If demand point i is assigned to warehouse v in the period t 1, otherwise 0
$x_{0 i}^{v t}$	If demand point i is the first demand point in any route of warehouse v in the period t
$x_{i l}^{v t}$	If demand point i is visited just after demand point i in any route of warehouse v in the period t
$x_{i 0}^{v t}$	If demand point i is the last demand point in any route of warehouse v in the period t
$u_{t i f}$	A continuous variable which represents the load of the vehicle after visiting demand point i in the period t
$r_{i v t}$	Response time to the demand point i in the period t if it is allocated to the warehouse v

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Sets
$V$	Set of indexes for warehouses $v \in V$
$T$	Set of indexes for periods $t \in T$
$I$	Set of indexes for demand points $i \in I$
$F$	Set of indexes for products $f \in F$
Parameters
$λ_{v t}$	The arrival rate of vehicles at the warehouse $v$ in the period $t$
$µ_{v t}$	Unloading rate at the warehouse $v$ at period $t$
$ρ_{v t}$	The traffic intensity of queue formed warehouse $v$ in the period $t$
$k_{v}$	Waiting rooms space for the warehouse $v$
$S_{v}$	Maximum number of servers at the warehouse $v$
${cap}_{v f}$	The capacity of arrival vehicles to the warehouse $v$ for product type f
$p_{i t f}$	The demand of demand point $i$ in the period t for product type f
$Q_{v f}$	The capacity of distribution vehicles of the warehouse $v$ for product type f
$τ_{i l}$	The traveling time between demand point $i$ and demand point $l$
$τ_{i v}^{'}$	The traveling time between demand point $i$ and warehouse $v$
$c_{i l}$	Traveling cost between demand point $i$ and $l$
$w t_{i}$	Importance of demand point $i$
$B_{i}$	Unloading time in the demand point $i$
$ω_{v f}$	Inventory cost of warehouse $v$ for product type $f$
Decision variables
$s_{v}$	Number of allocated servers to warehouse $v$
$w_{v t}$	Average waiting time for vehicles in the warehouse $v$ in the period $t$
$w q_{v t}$	Average waiting time for vehicles in the queues in the warehouse $v$ in the period $t$
$L_{v t}$	The average number of vehicles in the warehouse $v$ in the period $t$
$L q_{v t}$	The average number of vehicles in the queues in the warehouse $v$ in the period $t$
$π_{n}^{v t}$	Probability of existence of n vehicles in the warehouse v in the period t
$I_{v t f}$	Inventory level of product f in the warehouse v at the end of the period t
$z_{i v t}$	If demand point i is assigned to warehouse v in the period t 1, otherwise 0
$x_{0 i}^{v t}$	If demand point i is the first demand point in any route of warehouse v in the period t
$x_{i l}^{v t}$	If demand point i is visited just after demand point i in any route of warehouse v in the period t
$x_{i 0}^{v t}$	If demand point i is the last demand point in any route of warehouse v in the period t
$u_{t i f}$	A continuous variable which represents the load of the vehicle after visiting demand point i in the period t
$r_{i v t}$	Response time to the demand point i in the period t if it is allocated to the warehouse v

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