A Branch-and-Price Approach for a Ship Routing Problem ... - gerad

A Branch-and-Price Approach
for a Ship Routing Problem
with Multiple Products and
Inventory Constraints
Rutger de Mare 1
Dennis Huisman 2,3
1. ORTEC, Gouda
2. Econometric Institute and ECOPT, Erasmus Univ. Rotterdam
3. Dept. of Logistics, Netherlands Railways
1

Outline
• Practical Application
• Problem description
• Mathematical Formulation
• Column generation algorithm
2

Practical Application
Oil Company
• 1 refinery (no inventory constraints)
• ±12 depots (one per harbor)
• ± 8 grades
• ± 7 ships
• compartments (differ in size, 3-6)
15
10
40 30
10
3

15
10
40 30
10
4

Practical Application
• There is a schedule
• Question: Can we adjust the schedule due
to changes in the data?
• We construct an “optimal” schedule from
scratch
5

Problem Description
• Ships
– Compartments with capacities
– Travel times for each possible trip
– Fixed (un)loading times
– Wait at harbor is possible
• Harbor
– Inventory constraints
6

Inventory levels
Upper Alarm
Zone
Preferred
Inventory Zone
Lower Alarm
Zone
Unpumpables
Hard Maximal
Inventory Level
(tank top/capacity)
Soft Maximal
Inventory Level
(Alarm Level +)
Actual Inventory
Level
Soft Minimal
Inventory Level
(Alarm Level -)
Hard Minimal
Inventory Level
(Unpumpable Level)
0
Ullage
7

Inventory levels
8

Notation (Christiansen, 1998)
• (i,m): harbor i, number of arrival m
• (i,m,j,n): traveling from (i,m) to (j,n)
• V: set of ships
• H: set of harbors
• G: set of grades
• C: set of compartments
9

Decision Variables
• λ cr: compartment c travels schedule r
• θ igs: harbor i follows for grade g
sequence s
10

Mathematical Formulation (1)
min
∑∑
c∈Cr∈R c
Sailing cost route r
(only for c=1)
C
cr
∑∑ ∑
C λ + C θ
cr
i∈Hg∈G s∈S
S ig
HG
igs
Penalties for violating
alarm levels
igs
11

Mathematical Formulation (2)
c∈C
∑∑
v∈Vr∈R
∑
r∈R
c
∑
c
B
Q
imc
*
v
r
C
imcgr
λ
c
*
v
cr
r
+
−
∑
s∈S
∑
s∈S
ig
Y
Q
imgs
HG
imgs
igs
igs
= 1
=
0
∀i,
m,
1 if (i,m) is in schedule r to serve g 0 if (i,m) is in sequence s to (un)load g
λ
Quantity of g (un)loaded at (i.m)
in schedule r
θ
θ
∀i,
m,
g
g
Quantity of g (un)loaded at (i,m)
in sequence s
12

Mathematical Formulation (3)
∑∑
v∈Vr∈R
∑
r∈R
c
*
v
c
*
v
T
T
C
imc
C
imc
*
v
r
*
v
r
λ
λ
Time (un)loading starts at (i,m)
in schedule r, first compartment
c
*
v
r
c
*
v
r
−
Time (un)loading starts at (i,m)
in schedule r, first compartment
−
r∈R
s∈S
∑
∑
c
ig
T
T
C
imcr
HG
imgs
λ
θ
cr
igs
=
= 0
0
∀i,
m,
g
Time (un)loading starts at (i,m)
in sequence s
∀v,
i,
m,
c
Time (un)loading starts at (i,m)
in schedule r, compartment c
13

14
Mathematical Formulation (3)
)
,
,
,
(
,
}
1
,
0
{
,
,
0
,
0
,
1
1
n
j
m
i
c
X
s
g
i
r
c
g
i
c
c
ig
c
R
r
cr
imjncr
igs
cr
S
s
igs
R
r
cr
∀
∈
∀
≥
∀
≥
∀
=
∀
=
∑
∑
∑
∈
∈
∈
λ
θ
λ
θ
λ

Branch-and-Price Framework
Add initial columns to RMP
Solve LP relaxation of the RMP
Solve HGP for each HG Solve CP for each C
Find good columns
for each Harbour-Grade
Add new columns
to RMP
Good columns found?
Yes No
Find good columns
for each Compartment
Find integer
solution RMP
15

Compartment Sub problem
• Node: harbor arrival plus quantity per grade
• Reduced cost per arc:
– Fixed
– Time dependent
16

Compartment Sub problem
s
• Shortest path from s to any other node (taking
into account time aspect)
• Can be solved with Dynamic Programming
(Christiansen, 1998/9)
17

Branching
• Arcs in the network
• Aggressive 1-branch
18

Implementation Issues
• Pricing problem: exact or heuristically?
• Difficult to get a feasible solution of the
model: relax and penalize constraints?
• Cycles in the compartment sub problem
(generates redundant columns)
19

Results
• Will follow
20

Questions?
21