Providing a Mathematical Routing-Inventory Model for the Drug Supply Chain Considering the Travel Time Dependence and Perishability on Multiple Graphs

Table 1

Indices, variables, and parameters of the proposed model.

Indices and sets
	Index for nodes; node 0 represents the central warehouse and other nodes represent customers.
Pt	Index of periods of the planning horizon
vt	Index of vehicle types
P	Index of the p-th path between two nodes
r	The number of intervals into which the delivery time horizon is divided such that the travel speed on the route remains constant in each interval.
m	Index of intervals in which travel time is a piecewise linear function with a fixed slope in each interval.
Parameters
	The m-th interval of the travel time function
	The r-th interval of the travel speed function
	Travel speed on path p between nodes i and j for vehicle vt in interval r
	The distance between nodes i and j via path p
	The time that vehicle vt needs to depart from node i toward node j in order to arrive at its destination at a specified time via path p in interval r
	The time that vehicle vt arrives at node j via path p if it departs from node i in interval m.
	Travel time from node i to node j via path p for vehicle vt if it departs from node i in interval m
	The slope of the piecewise linear function of travel time for the node pair (i, j), vehicle vt, path p, and interval m
	The intercept of the piecewise linear function of travel time for the node pair (i, j), vehicle vt, path p, and interval m
	Service time for customer i
	The demand of customer i in period pt
	The capacity of a vehicle vt
	Warehousing capacity of node i
	Fixed cost of using vehicle vt
	The time-dependent variable of using vehicle vt
	Cost of storing the product in node i for one period
	The unit price of the product
	Product spoilage rate when stored in node i
	Initial inventory level of node i at the beginning of the planning horizon (it is assumed that the initial inventory level of the central warehouse is 0).
	Number of vehicles of type vt
	Length of the delivery time horizon in each period of the planning horizon
Variables
Z	Objective function value
	=1 if the edge connecting node i to node j is selected as the optimal route for vehicle vt in period pt; = 0 otherwise
	=1 if the edge connecting node i to node j via path p is selected as the optimal route for vehicle vt in period pt;=0 otherwise
	=1 if the edge connecting node i to node j via path p is selected as the optimal route for vehicle vt in interval m in period pt; = 0 otherwise
	Time of departure of vehicle vt from node i toward node j via path p in interval m in period pt
	The amount of product transported on the activated edge connecting node i to node j in period pt
	The amount of product transported on the activated edge connecting node i to node j via path p in period pt
	The amount of product transported by vehicle vt on the activated edge connecting node i to node j via path p in period pt
	Inventory level of node i at the end of period pt
	The amount of product spoiled in the warehouse of node i during the period pt
	The amount of product delivered to customer i in period pt
	The amount of product arriving at the central warehouse during the period pt