Information Technology Reference
In-Depth Information
2 Mathematical Formulation
For modeling the problem we define the following notation.
Sets
T
Set of trailers.
T
i
Subset of trailers that are initially located at node
i
.
T
e
(
T
w
) Subset of trailers with (without) a shelf.
K
Set of vehicles.
K
i
Subset of vehicles that are initially located at node
i
.
V
Plants and distribution centers (nodes).
E
Edges.
P
Products.
P
R
(
P
N
) Subset of returnable (non-returnable) products.
H
Time index (usually hours) set.
Parameters
c
top
t
Capacity (number of pallets) on top compartment of trailer
t
.
c
bo
t
Capacity (number of pallets) on bottom compartment of trailer
t
.
τ
t
Time in which trailer
t
can start its route.
τ
t
Time in which trailer
t
must return to its plant.
σ
(
t
) Original location of trailer
t
.
σ
(
k
) Original location of vehicle
k
.
n
ip
Amount of product
p
(number of 12-bottle boxes) that must be picked
up from node
i
.
n
ip
Amount of product
p
that must be delivered to node
i
.
(
a
i
,b
i
) Time window for servicing node
i
.
o
p
Number of 12-bottle boxes of product
p
that fit in a pallet.
c
ij
Travel cost from node
i
to
j
using a single vehicle.
c
ij
Travel cost from node
i
to
j
using a double vehicle.
s
ij
Travel time from node
i
to
j
using a single vehicle.
s
ij
Travel time from node
i
to
j
using a double vehicle.
S
Customer service time at each node.
f
ih
Available dock capacity in node
i
at time
h
.
C
Fixed cost for vehicle use.
Binary variables
w
kt
= 1, if trailer
t
is assigned to vehicle
k
;0,otherwise
x
ijk
= 1, if vehicle
k
travels directly from node
i
to
j
;0,otherwise
y
ikh
= 1, if vehicle
k
arrives at node
i
at time
h
;0,otherwise
z
k
= 1, if vehicle
k
is used; 0, otherwise
Integer variables
v
top
k
Number of pallets on top of vehicle
k
.
v
bel
k
Number of pallets on bottom of vehicle
k
.
