A GRASP for a Multi-depot Multi-commodity Pickup and Delivery Problem with Time Windows and Heterogeneous Fleet in the Bottled Beverage Industry - Computational Logistics - page 146

v typ k Configuration of vehicle k (=-1/0/1 if unassigned/single/double).
g ikp Amount of product p (number of 12-bottle boxes) that vehicle k will deliver
at node i .
g ikp Amount of product p that vehicle k will pickup at node i .
q ikp Amount of product p in vehicle k after servicing node i .
Note: At the start of a route, g σ ( k ) kp =0and q σ ( k ) kp = g σ ( k ) kp ,for k

∈ K,p ∈ P .

Real variables
( l k ,u k ) Starting and finishing time of vehicle k .
s ik Starting service time of vehicle k at node i .
c ik Accumulated cost of vehicle k up to node i .
OC k Total travel cost for vehicle k .
OT k Total travel time for vehicle k .
Model

Minimize
k

OC k +
k

Cz k (1)

∈ K ∈ K

subject to
t

w kt ≤ k ∈ K

2 (2)

∈ T σ ( k )

w kt ≤ t ∈ T

1 (3)

k ∈ K σ ( t )

=
t

v type
k

w kt − k ∈ K

1 (4)

∈ T σ ( k )

u k ≤ τ t + M T (1 − w kt )

k ∈ K,t ∈ T σ ( k ) (5)

τ t w kt l k ≥ k ∈ K,t ∈ T σ ( k ) (6)

v top
k

c to t w kt ≤ k ∈ K (7)

t∈T σ ( k )

v bel
k

c be t w kt ≤ k ∈ K (8)

t∈T σ ( k )

x ijk ≤ z k i,j ∈ V,k ∈ K (9)

w kt ≥ z k k ∈ K (10)

t ∈ T σ ( k )

x ijk − x jik =0 j ∈ V,k ∈ K (11)

i :( i,j ) ∈E i :( j,i ) ∈E

x ijk ≤ 1 j ∈ V,k ∈ K (12)

i :( i,j ) ∈ E

a i
j :( j,i ) ∈E

b i
j :( j,i ) ∈E

x jik ≤ s ik ≤ x jik i ∈ V,k ∈ K (13)

a σ ( k ) z k ≤ OT k ≤ b σ ( k ) z k k ∈ K (14)

l k − M T (1 − z k ) ≤ s σ ( k ) k ≤ u k + M T (1 − z k ) k ∈ K (15)