Multidepot Recycling Vehicle Routing Problem with Resource Sharing and Time Window Assignment

Table 2

Notations and definitions in the MRVRPRSTWA.

Set definition
P	Set of recycling centers (RCs), P = {p|p = 1, 2, 3…, m} and m is the total number of RCs
C	Set of all recycling customers, C = {c|c = 1, 2, 3…, h} and h is the total number of customers
Cf	Set of customers with fixed time windows
Ca	Set of customers with assigned time windows
V	Set of vehicles for recycling, V = {| = 1, 2, 3…, b} and b is the total number of vehicles
S	Set of semitrailer trucks used between RCs, S = {s|s = 1, 2, 3…, w} and w is the total number of vehicles
	Maximum times for executing recycling routes of vehicle in a working period, ∈V
Parameters
qi	Recycling demand quantity of customer i, i∈C
tqmn	Transportation quantity from logistics facility m to n, m, n∈P
Qs	Maximum capacity of semitrailer truck s, s∈S
	Maximum capacity of vehicle , ∈V
Qm	Maximum capacity of logistics facility m, m∈P
Us	Usage cost of semitrailer truck s, s∈S (unit: dollar/km)
	Usage cost of vehicle , ∈V (unit: dollar/km)
Dij	Distance from logistics facility or customer i to j, i, j∈P∪C, i ≠ j (unit: km)
Dmn	Distance from logistics facility m to logistics facility n, m, n∈P (unit: km)
W	Number of working days in one planning period
B	Number of planning periods in one year
Ms	Annual MC of semitrailer truck s, s∈S
	Annual MC of vehicle , ∈V
FLm	Fixed cost of logistics facility m, m∈P
[ei,ti]	Fixed time window of customer i, i∈Cf
[e’ i,l’ i]	Expected time window of customer i, i∈Ca
[αi,βi]	Time window assigned to customer i, i∈Ca
[Em,Lm]	Service time window of logistics facility m, m∈P
μe	Penalty cost for early arrival per unit time
μd	Penalty cost for delayed arrival per unit time
μa	Cost coefficient of the customer’s time window from the expected time window to the assigned time window per unit time
dtk vm	Departure time of the kth route of vehicle V from logistics facility m, ∈ V, m ∈ P, k ∈
atk iv	Arrival time of the kth route of vehicle v at node i, ∈ V, i ∈ P ∪ C, k ∈
tijv	Travel time of vehicle between entities i and j, i, j ∈ P ∪ C, ∈ V
|Ns|	Total number of logistic facilities served by semitrailer truck s, s∈S
	Total number of customers served by vehicle , ∈V
BN	Big number
Decision variables
xijv	If vehicle travels from logistics facility or customer i to j, then xijv = 1; otherwise, xijv = 0, i, j ∈ P∪C, ∈ V
zk v	If vehicle has the kth route, then zk v = 1; otherwise, zk v = 0, m ∈ P, ∈ V, and k ∈
ωivm	If vehicle departs from logistics facility m to served customer i, ωivm = 1; otherwise, ωivm = 0, i ∈ C, ∈ V, and m ∈ P
ymns	If semitrailer truck s transports between logistics facility m and n, ymnk = 1; otherwise, ymns = 0, m, n ∈ P, and s ∈ S
τimn	If customer i’s logistic facility changes from logistics facility m to n after optimization, τimn = 1; otherwise, τimn = 0, i ∈ C, and m, n ∈ P
	If semitrailer truck s departs from logistics facility m,  = 1; otherwise,  = 0, s ∈ S, and m ∈ P