Resource Constrained Shortest Paths with
Side Constraints and Non Linear Costs
Stefano Gualandi
[email protected]
http://www-dimat.unipv.it/~gualandi
twitter: @famo2spaghi
Thursday, September 19, 13
Outline
1. Introduction
2. Non Linear Costs
3. Iterated Preprocessing
4. Lower bounding via Lagrangian Relaxation
5. Computational Results
Thursday, September 19, 13
Column
GenerationPersepective
Overview
Column Generation:
Algorithmic
What is F in Crew Scheduling problems?
Thursday, September 19, 13
Constrained Shortest Path
Column orResource
Variable Generation
The problem of putting together a set of pieces of work into a
single duty, that is a column or variable of problem (LP-MP), is
formalized as a
Resource Constrained Shortest Path Problem
Example 12 pieces of works, 3 depots
Thursday, September 19, 13
Introduction
Regional Transit
Resource Constraint Shortest Path
Resource Constrained Shortest Path
Resource Constraint Shortest Path
Let G = (N, A) be the compatibility graph, weighted, directed, and
acyclic:
N = P fi {{s h , t h }|h œ D} a node for each PoW, and a pair of
nodes for each depot
A has an arc for each pair (i, j) of compatible PoW, and (s h , i)
(pull-out) and (i, t h ) (pull-in) ’h œ D and i œ P
Thursday, September 19, 13
N = P fiResource
{{s h , t h }|h œConstrained
D}
Shortest Path
A
has
an arcoffor
eachtogether
pair (i, aj)set
of ofcompatible
PoW,
(s h , i)
The
problem
putting
pieces of work
into and
a single
duty, that is and
a column
variable of’h
problem
(LP-MP),
(pull-out)
(i, t hor
) (pull-in)
œ D and
i œ Pis formalized as a
k
each arc
(i,
j)
has
associated
a
set
of
resources
r
ij , for each k œ K ,
Resource Constrained Shortest Path Problem
e.g. working time, driving time, and break time (other resources
may
be used
to model
workingonregulation)
Example:
A possible
path/duty
G
Tuesday, June 4, 13
Thursday, September 19, 13
Example
of Crew
Schedules
(with reosurces)
Example
of Crew
Schedule
(Resources)
Resources:
1 spread time (red)
2 driving time (light blue), corresponds to PoW
Tuesday, June 4, 13
3 out-of-service time (yellow)
4 long break (grey)
5 breaks (green), very important how they are located
Thursday, September 19, 13
Non Linear Costs
Thursday, September 19, 13
Non Linear Costs
Thursday, September 19, 13
600
400
0
200
f(t(P)) = step(t(P)) + (t(P))^2
800
Non Linear Costs
0
60
120
180
240
300
360
Time
Thursday, September 19, 13
420
480
540
600
660
720
Lagrangian
Relaxation:
Arc-Flow
Formulation
Resource
Constrained
Shortest
Paths (RCSP)
3 4
Arc-flow IP formulation with non linear costs fh · :
min
s.t.
ÿ
eœA
ÿ
w e xe +
eœ”i+
ÿ
eœA
xe ≠
ÿ ÿ
kœK hœH
ÿ
eœ”i≠
rek xe Æ U k
fh
3ÿ
eœA
xe = bi =
rek xe
4
Y
_
] +1 if i = s
≠1 if i = t
_
[ 0
otherwise
’i œ N
’k œ K
+ side non linear constraints
xe œ {0, 1}
Thursday, September 19, 13
’e œ A.
Resource Constrained Shortest Paths (RCSP)
w e, t e
a
5, 1
5, 1
s
5, 1
5, 1
i
10, 0
Thursday, September 19, 13
b
5, 0
c
t
10, 0
5, 0
d
Definition
(Path Super
Additivity)
Resource Constrained
Shortest
Paths
(RCSP)
A (path) cost function is super additive iff:
We restrict to super additive functions: c(P1 fi P2 ) Ø c(P1 ) + c(P2 )
We consider here a specific type of super additive
3
c(P) = w (P) + f t(P)
w e, t e
a
5, 1
s
10, 0
Thursday, September 19, 13
b
5, 1
5, 1
!i "
=
ÿ
eœP
w5,e 1+ f
4
3ÿ
eœP
t
te
4
where f · is a super additive function. Since w (
c(P) is also super
additive.
5, 0
10, 0
5, 0
c
d
we,$te$
a
b
Additivity)
5,1 (Path Super
5,1
5,1
5,1 Definition
Resource
Constrained
Shortest
Paths
(RCSP)
A (path)
cost function is super additive
iff:
i
t
s
c(P1 fi P5,02 ) Ø c(P1 ) + c(P2 )
10,0
10,0to super
5,0
We restrict
additive
functions:
c
d
! " q
!q
"2
Example:
Consider c(P) = w (P)
f P =
we + typeeœP
Example:
We +
consider
here eœP
a specific
of tsuper
additive
e
There are 4 paths:
3
4
! "
c(P)
P1 = {s, a, i, b, t}, w (P1 ) = 20, f P1 = 16,
c(P1=
) = w36(P) + f t(P)
! "
w e, t e
3ÿ 4
a w (P2 ) = 25, f P2 = 4, c(P2 )b = 29
ÿ
P2 = {s, c, i, b, t},
5, 1
=
w5,e 1+ f
te
5, 1
! "5, 1
P3 = {s, a, i, d, t}, w (P3 ) = 25, f P3 = 4, c(P3 ) = 29
eœP
eœP
s
t
!i! " "
P4 = {s, c, i, d, t}, w (P4 )where
= 30,ff ·P4is=a 0,
c(P
)
=
30
4
super additive
function. Since w (
c(P) is also super
additive.
5, 0
10, 0
5, 0
10, 0
Thursday, September 19, 13
c
d
we,$te$
a
b
Additivity)
5,1 (Path Super
5,1
5,1
5,1 Definition
Resource
Constrained
Shortest
Paths
(RCSP)
A (path)
cost function is super additive
iff:
i
t
s
c(P1 fi P5,02 ) Ø c(P1 ) + c(P2 )
10,0
10,0to super
5,0
We restrict
additive
functions:
c
d
! " q
!q
"2
Example:
Consider c(P) = w (P)
f P =
we + typeeœP
Example:
We +
consider
here eœP
a specific
of tsuper
additive
e
There are 4 paths:
3
4
! "
c(P)
P1 = {s, a, i, b, t}, w (P1 ) = 20, f P1 = 16,
c(P1=
) = w36(P) + f t(P)
! "
w e, t e
3ÿ 4
a w (P2 ) = 25, f P2 = 4, c(P2 )b = 29
ÿ
P2 = {s, c, i, b, t},
5, 1
=
w5,e 1+ f
te
5, 1
! "5, 1
P3 = {s, a, i, d, t}, w (P3 ) = 25, f P3 = 4, c(P3 ) = 29
eœP
eœP
s
t
!i! " "
P4 = {s, c, i, d, t}, w (P4 )where
= 30,ff ·P4is=a 0,
c(P
)
=
30
4
super additive
function. Since w (
c(P) is also super
additive.
5, 0
10, 0
5, 0
10, 0
c
d
Optimal Path: P2 = {s, c, i, b, t}, c(P2) = 25 + 4 = 29
Thursday, September 19, 13
we,$te$
a
b
Additivity)
5,1 (Path Super
5,1
5,1
5,1 Definition
Resource
Constrained
Shortest
Paths
(RCSP)
A (path)
cost function is super additive
iff:
i
t
s
c(P1 fi P5,02 ) Ø c(P1 ) + c(P2 )
10,0
10,0to super
5,0
We restrict
additive
functions:
c
d
! " q
!q
"2
Example:
Consider c(P) = w (P)
f P =
we + typeeœP
Example:
We +
consider
here eœP
a specific
of tsuper
additive
e
There are 4 paths:
3
4
! "
c(P)
P1 = {s, a, i, b, t}, w (P1 ) = 20, f P1 = 16,
c(P1=
) = w36(P) + f t(P)
! "
w e, t e
3ÿ 4
14
a w (P2 ) = 25, f P2 = 4, c(P2 )b = 29
ÿ
P2 = {s, c, i, b, t},
5, 1
=
w5,e 1+ f
te
5, 1
! "5, 1
P3 = {s, a, i, d, t}, w (P3 ) = 25, f P3 = 4, c(P3 ) = 29
eœP
eœP
s
t
!i! " "
P4 = {s, c, i, d, t}, w (P4 )where
= 30,ff ·P4is=a 0,
c(P
)
=
30
4
super additive
function. Since w (
c(P) is also super
additive.
5, 0
10, 0
5, 0
10, 0
c
15
d
Optimal Path: P2 = {s, c, i, b, t}, c(P2) = 25 + 4 = 29
Thursday, September 19, 13
we,$te$
a
b
Additivity)
5,1 (Path Super
5,1
5,1
5,1 Definition
Resource
Constrained
Shortest
Paths
(RCSP)
A (path)
cost function is super additive
iff:
i
t
s
c(P1 fi P5,02 ) Ø c(P1 ) + c(P2 )
10,0
10,0to super
5,0
We restrict
additive
functions:
c
d
! " q
!q
"2
Example:
Consider c(P) = w (P)
f P =
we + typeeœP
Example:
We +
consider
here eœP
a specific
of tsuper
additive
e
There are 4 paths:
3
4
! "
c(P)
P1 = {s, a, i, b, t}, w (P1 ) = 20, f P1 = 16,
c(P1=
) = w36(P) + f t(P)
! "
w e, t e
3ÿ 4
14
a w (P2 ) = 25,
14f P2 = 4, c(P2 )b = 29
ÿ
P2 = {s, c, i, b, t},
5, 1
=
w5,e 1+ f
te
5, 1
! "5, 1
P3 = {s, a, i, d, t}, w (P3 ) = 25, f P3 = 4, c(P3 ) = 29
eœP
eœP
s
t
!i! " "
P4 = {s, c, i, d, t}, w (P4 )where
= 30,ff ·P4is=a 0,
c(P
)
=
30
4
super additive
function. Since w (
c(P) is also super
additive.
5, 0
10, 0
5, 0
10, 0
c
15
d
Optimal Path: P2 = {s, c, i, b, t}, c(P2) = 25 + 4 = 29
Thursday, September 19, 13
we,$te$
a
b
Additivity)
5,1 (Path Super
5,1
5,1
5,1 Definition
Resource
Constrained
Shortest
Paths
(RCSP)
A (path)
cost function is super additive
iff:
i
t
s
c(P1 fi P5,02 ) Ø c(P1 ) + c(P2 )
10,0
10,0to super
5,0
We restrict
additive
functions:
c
d
! " q
!q
"2
Example:
Consider c(P) = w (P)
f P =
we + typeeœP
Example:
We +
consider
here eœP
a specific
of tsuper
additive
e
There are 4 paths:
3
4
! "
c(P)
P1 = {s, a, i, b, t}, w (P1 ) = 20, f P1 = 16,
c(P1=
) = w36(P) + f t(P)
! "
w e, t e
3ÿ 4
36
a w (P2 ) = 25, f P2 = 4, c(P2 )b = 29
ÿ
P2 = {s, c, i, b, t},
5, 1
=
w5,e 1+ f
te
5, 1
! "5, 1
P3 = {s, a, i, d, t}, w (P3 ) = 25, f P3 = 4, c(P3 ) = 29
eœP
eœP
s
t
!i! " "
P4 = {s, c, i, d, t}, w (P4 )where
= 30,ff ·P4is=a 0,
c(P
)
=
30
4
super additive
function. Since w (
c(P) is also super
additive.
5, 0
10, 0
5, 0
10, 0
c
d
Path: P3 = {s, a, i, b, t}, c(P2) = 20 + 16 = 36
Thursday, September 19, 13
we,$te$
a
b
Additivity)
5,1 (Path Super
5,1
5,1
5,1 Definition
Resource
Constrained
Shortest
Paths
(RCSP)
A (path)
cost function is super additive
iff:
i
t
s
c(P1 fi P5,02 ) Ø c(P1 ) + c(P2 )
10,0
10,0to super
5,0
We restrict
additive
functions:
c
d
! " q
!q
"2
Example:
Consider c(P) = w (P)
f P =
we + typeeœP
Example:
We +
consider
here eœP
a specific
of tsuper
additive
e
There are 4 paths:
3
4
! "
c(P)
P1 = {s, a, i, b, t}, w (P1 ) = 20, f P1 = 16,
c(P1=
) = w36(P) + f t(P)
! "
w e, t e
3ÿ 4
a w (P2 ) = 25,
14f P2 = 4, c(P2 )b = 29
ÿ
P2 = {s, c, i, b, t},
5, 1
=
w5,e 1+ f
te
5, 1
! "5, 1
P3 = {s, a, i, d, t}, w (P3 ) = 25, f P3 = 4, c(P3 ) = 29
eœP
eœP
s
t
!i! " "
P4 = {s, c, i, d, t}, w (P4 )where
= 30,ff ·P4is=a 0,
c(P
)
=
30
4
super additive
function. Since w (
c(P) is also super
additive.
5, 0
10, 0
5, 0
10, 0
c
15
d
Bellmann’s optimality conditions do not hold!
Thursday, September 19, 13
Outline
1. Introduction
2. Non Linear Costs
3. Iterated Preprocessing
4. Lower bounding via Lagrangian Relaxation
5. Computational Results
Thursday, September 19, 13
Resource-basedResource-based
Filtering
Preprocessing
⌫

(Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007)
if r k (Psiú ) + rek + r k (Pjtú ) > U k then remove arc e = (i, j)
where Psiú and Pjtú are shortest (k-th resource) paths.
Resource consumption of each arc. Upper resource bound U = 7.
6
a
b
1
2
2
3
s
1
4
2
c
Thursday, September 19, 13
t
1
d
Resource-basedResource-based
Filtering
Preprocessing
⌫

(Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007)
if r k (Psiú ) + rek + r k (Pjtú ) > U k then remove arc e = (i, j)
where Psiú and Pjtú are shortest (k-th resource) paths.
Resource consumption of each arc. Upper resource bound U = 7.
6
a
re
a
2 2
s
3
3
s
4
4
b
b
2
2
1
1
1
1
t
2
c
c
Thursday, September 19, 13
6
d
1
1
d
2
t
Resource-basedResource-based
Filtering
Preprocessing
⌫

(Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007)
if r k (Psiú ) + rek + r k (Pjtú ) > U k then remove arc e = (i, j)
where Psiú and Pjtú are shortest (k-th resource) paths.
Resource consumption of each arc. Upper resource bound U = 7.
6
a
re
a
2 2
s
3
3
s
4
4
b
b
2
2
1
1
1
1
t
2
c
c
Thursday, September 19, 13
6
d
1
1
d
2
t
Resource-basedResource-based
Filtering
Preprocessing
⌫

(Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007)
if r k (Psiú ) + rek + r k (Pjtú ) > U k then remove arc e = (i, j)
where Psiú and Pjtú are shortest (k-th resource) paths.
Resource consumption of each arc. Upper resource bound U = 7.
6
a
re
a
2 2
s
3
3
s
4
4
b
b
2
2
1
1
1
1
t
2
c
c
Thursday, September 19, 13
6
d
1
1
d
2
t
Resource-basedResource-based
Filtering
Preprocessing
⌫

(Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007)
if r k (Psiú ) + rek + r k (Pjtú ) > U k then remove arc e = (i, j)
where Psiú and Pjtú are shortest (k-th resource) paths.
Resource consumption of each arc. Upper resource bound U = 7.
6
a
re
a
2 2
s
3
3
s
4
4
b
b
2
2
1
1
1
1
t
2
c
c
Thursday, September 19, 13
6
d
1
1
d
2
t
Resource-basedResource-based
Filtering
Preprocessing
⌫

(Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007)
if r k (Psiú ) + rek + r k (Pjtú ) > U k then remove arc e = (i, j)
where Psiú and Pjtú are shortest (k-th resource) paths.
Resource consumption of each arc. Upper resource bound U = 7.
6
a
re
a
2 2
s
3
3
s
4
4
b
b
2
2
1
1
1
1
t
2
c
c
Thursday, September 19, 13
6
d
1
1
d
2
t
Cost-based Preprocessing
UB=7
ce
2
a
3
4
b
1
s
Thursday, September 19, 13
6
2
1
t
2
c
1
d
Cost-based Preprocessing
UB=7
ce
2
a
3
4
b
1
s
Thursday, September 19, 13
6
2
1
t
2
c
1
d
Cost-based Preprocessing
⌫

if LB(c(P ú e )) Ø UB then remove arc e
s≠
æt
ú
where P e is a shortest path from s to t via arc e.
s≠
æt
There are at least three methods to compute such lower bound
(see our poster!)
UB=7
ce
6
a
b
1
The most2 effective is based on a Lagrangian Relaxation
s
3
4
Thursday, September 19, 13
2
1
t
2
c
1
d
Outline
1. Introduction
2. Non Linear Costs
3. Iterated Preprocessing
4. Lower bounding via Lagrangian Relaxation
5. Computational Results
Thursday, September 19, 13
Lagrangian Relaxation: Arc-Flow Formulation
Resource Constrained Shortest Paths (RCSP)
! "
Arc-flow IP formulation with non linear costs f ·
ÿ
min
eœA
ÿ
s.t.
w e xe +
eœ”i+
ÿ
eœA
xe ≠
ÿ ÿ
kœK hœH
ÿ
eœ”i≠
rek xe Æ U k
fh
3ÿ
eœA
xe = bi =
rek xe
4
h:
Y
_
] +1 if i = s
≠1 if i = t
_
[ 0
otherwise
’i œ N
’k œ K
+ side non linear constraints
xe œ {0, 1}
Thursday, September 19, 13
’e œ A.
Lagrangian Relaxation: Arc-Flow Formulation
Resource Constrained Shortest Paths (RCSP)
! "
Arc-flow IP formulation with non linear costs f · :
min
ÿ
w e xe + f
eœA
ÿ
s.t.
eœ”i+
ÿ
eœA
xe ≠
3ÿ
ÿ
eœ”i≠
rek xe Æ U k
eœA
re1 xe
4
xe = bi =
Y
_
] +1 if i = s
≠1 if i = t
_
[ 0
otherwise
xe œ {0, 1}
[G. Tsaggouris and C. Zaroliagis, ESA2004]
Thursday, September 19, 13
’i œ N
’k œ K
’e œ A.
Resource
Constrained
Shortest
Paths (RCSP)
Lagrangian
Relaxation:
Arc-Flow
Formulation
! "
Arc-flow IP formulation with non linear costs f · :
min
s.t.
ÿ
eœA
ÿ
3 4
w e xe + f z
eœ”i+
ÿ
eœA
ÿ
eœA
xe ≠
ÿ
eœ”i≠
rek xe Æ U k
≠1 if i = t
_
[ 0
otherwise
’i œ N
’k œ K
re1 xe = z
xe œ {0, 1}
Thursday, September 19, 13
xe = bi =
Y
_
] +1 if i = s
’e œ A.
Lower Bounding: Lagrangian Relaxation
The arc-flow
LP relaxation of RCSP with a super additive cost
! "
function f · is:
min
s.t.
ÿ
eœA
w e xe + f z
ÿ
eœ”i+
multiplier –k Æ 0
multiplier — Æ 0
æ
æ
ÿ
eœA
ÿ
eœA
xe ≠
ÿ
eœ”i≠
rek xe Æ U k
xe = bi
(8)
’i œ N
(9)
’k œ K
(10)
t e xe Æ z
xe Ø 0
Thursday, September 19, 13
! "
(11)
’e œ A.
(12)
LagrangianLower
Relaxation:
Bounding:Arc-Flow
LagrangianFormulation
Relaxation
Iti is possible to formulate the following Lagrangian dual:
ÿ
(–, —) = ≠
–k U k +
kœK
+ min
s.t.
ÿ
eœ”i+
ÿ
eœA
xe ≠
xe Ø 0
A
we +
ÿ
eœ”i≠
ÿ
kœK
xe = bi
–k rek
+ —te
B
’i œ N
! "
xe + f z ≠—z
’e œ A.
This problem decomposes into two subproblems and is solved via a
subgradient optimization algorithm:
1
The x variables define a shortest path problem
2
The z variable defines an unconstrained optimization problem
Thursday, September 19, 13
⌫
Cost-based
Preprocessing via Lagrangian Lower Bounds
if LB(c(P ú e )) Ø UB then remove arc e
s≠
æt
ú
where P e is a shortest path from s to t via arc e.
⌫
s≠
æt

if LB(c(P ú e )) Ø UB then remove arc e
s≠
æt
ú
There where
are at Pleast
three
methods
to compute
is a shortest
path
from s tosuch
t vialower
arc e.bound
e
s≠
æt
(see our poster!)
The are
most
is based
on to
a Lagrangian
Relaxation
There
at effective
least three
methods
compute such
lower bound
(see our poster!)
⌫

! "
ú
ú
¯
c(P
e ) Ø w̄ (P e ) + min{f z ≠ —z}
The most effective
is based
Relaxation
s≠
æt
s≠
æton a Lagrangian
q
¯ e]
[with reduced costs w̄e = we + kœK –
¯ k rek + —t
Thursday, September 19, 13
Filter and Dive
Algorithm 1: FilterAndDive(G, LB, U B, F g , B g , U g )
1
2
3
4
5
6
7
Input: G = (N, A) directed graph and distance function g(·)
Input: (LB, U B) lower and upper bounds on the optimal path
Input: F g , B g forward and backward shortest path tree as function of g(·)
Input: U g upper bound on the path length as function of g(·)
Output: An optimum path, or updated U B, or a reduced graph
foreach i 2 N do
if Fig + Big > U g then
N
N \ {i}
FILTER...
else
foreach e = (i, j) 2 A do
if Fig + g(e) + Bjg > U g then
A
A \ {e}
8
9
10
11
12
13
14
15
Thursday, September 19, 13
...DIVE
else
if PathCost(Fig , e, Bjg ) < U B^ PathFeasible(Fig , e, Bjg ) then
⇤
Pst
MakePath(Fig , e, Bjg );
⇤
Update U B and store Pst
;
check for side
if LB U B then
constraints
⇤
return Pst (that is an optimum path)
else
A
A \ {e}
Closing the Duality Gap
Near Shortest Path Enumeration
After reaching a fixpoint, if LB < UB then, we apply a near
shortest path enumeration algorithm (Carlyle et al., 2008).
We compute shortest reversed distances for every resource and for
reduced costs
Then we perform a depth-first search from s. When a vertex i is
visited, the algorithm backtracks if
1
for any resource k, the consumption of Psi plus the reversed
(resource) distance to t exceeds U k
2
the reduced cost of Psi plus the reversed (reduced cost)
distance to t exceeds UB
3
the cost c(Psi ) Ø UB
Thursday, September 19, 13
Outline
1. Introduction
2. Non Linear Costs
3. Iterated Preprocessing
4. Lower bounding via Lagrangian Relaxation
5. Computational Results
Thursday, September 19, 13
Constrained Path Solver: Scalability
Thursday, September 19, 13
Computational Results: Stepwise Function
Resource and Cost-based Preprocessing
Non linear costs: extra allowances
Each row gives the averages over 16 instances, with 7 resources.
is percentage of removed arcs
Gap is
UB≠Opt
Opt
◊ 100
Graphs
Resource
Reduced Cost Exact
n
m
Time
Time
Gap Time
4137 135506 0.77 22.5% 3.12 30.2% 0.0% 75.1
2835 132468 0.59 40.3% 2.35 45.4% 0.0% 30.6
3792 134701 0.92 30.2% 2.87 37.4% 0.0% 69.3
Thursday, September 19, 13
Crew Scheduling: Real Life Instances
Instance
158TG
171TG
182TG
217TG
233TG
254TG
274TG
300TG
425TG
560TG
Pieces
684
802
846
967
1067
1169
1240
1369
1865
2314
Glob. Const.
4
6
7
8
8
10
11
12
32
21
Depots
3
6
7
8
8
10
11
12
16
10
Non linear component: step-wise on single resource
Side constraints: non linear constraint on break distribution
Thursday, September 19, 13
Logo vettoriale MAIOR Orizzontale
Logo vettoriale MAIOR Verticale
m. (%)
costi
sol.
e
tempi
Impact on Column Generation-based Heuristic
Miglioram costo (%)
1.00%
0.80%
0.60%
0.40%
0.20%
0.00%
158TG
171TG
182TG
217TG
233TG
254TG
274TG
300TG
Logo vettoriale MAIOR Orizzontale
425TG
-0.20%
-0.40%
Logo vettoriale MAIOR Verticale
Miglioram Tempo (%)
Labeling-heuristic vs. Exact New Algorithm
100.00%
Difference of cost solution obtained via column generation
90.00%
80.00%
Thursday, September 19, 13
560TG
-0.40%
Impact on Column Generation-based Heuristic
Miglioram Tempo (%)
100.00%
90.00%
80.00%
70.00%
60.00%
50.00%
40.00%
30.00%
Logo vettoriale MAIOR Orizzontale
20.00%
10.00%
0.00%
158TG
171TG
182TG
217TG
233TG
254TG
274TG
300TG
Labeling-heuristic vs. Exact New Algorithm
Difference of run time obtained via column generation
Thursday, September 19, 13
425TG
Logo vettoriale MAIOR Verticale
560TG
5