S IMULTANEOUS C OLUMN - AND -ROW G ENERATION
FOR S OLVING
L ARGE -S CALE L INEAR P ROGRAMS WITH
C OLUMN -D EPENDENT-ROWS
İbrahim Muter
CIRRELT and HEC Montreal
with
Ş. İlker Birbil and Kerem Bülbül
Sabancı University, Istanbul, TURKEY
2012
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
1 / 33
I NTRODUCTION
Main Motivation
Contributions
C OLUMN -D EPENDENT-ROWS P ROBLEMS
Generic Mathematical Model
Assumptions
P ROPOSED S OLUTION M ETHOD
y-pricing subproblem (y-PSP)
x-pricing subproblem (x-PSP)
Row-Generating Pricing Subproblem
C ONCLUSIONS AND F UTURE R ESEARCH D IRECTIONS
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
2 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN - AND -ROW G ENERATION
minimize
...
subject to
=
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
3 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN - AND -ROW G ENERATION
minimize
...
subject to
=
A subset of all columns
A subset
of all
constraints
S RMP
(Short RMP)
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
3 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN - AND -ROW G ENERATION
minimize
...
subject to
=
A subset of all columns
A subset
of all
constraints
Optimal
duals
S RMP
(Short RMP)
İbrahim Muter (CIRRELT)
Pricing
subproblem
Simultaneous Column-and-Row Generation
2012
3 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN - AND -ROW G ENERATION
minimize
...
subject to
=
A subset of all columns
A subset
of all
constraints
Optimal
duals
New SRMP
S RMP
(Short RMP)
İbrahim Muter (CIRRELT)
Column(s) with
negative reduced cost(s)?
Pricing
subproblem
Simultaneous Column-and-Row Generation
New linking constraints
2012
3 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN - AND -ROW G ENERATION
minimize
...
subject to
=
Duals of the
missing constraints?
A subset of all columns
A subset
of all
constraints
Optimal
duals
New SRMP
S RMP
(Short RMP)
İbrahim Muter (CIRRELT)
Column(s) with
negative reduced cost(s)?
Pricing
subproblem
Simultaneous Column-and-Row Generation
New linking constraints
2012
3 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN -D EPENDENT ROWS
(CDR) P ROBLEMS
Duals of the
missing constraints?
A subset of all columns
A subset
of all
constraints
Optimal
duals
New SRMP
S RMP
(Short RMP)
İbrahim Muter (CIRRELT)
Column(s) with
negative reduced cost(s)?
Pricing
subproblem
Simultaneous Column-and-Row Generation
New linking constraints
2012
4 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN -D EPENDENT ROWS
(CDR) P ROBLEMS
Duals of the
missing constraints?
A subset of all columns
A subset
of all
constraints
I
Optimal
duals
Column(s) with
negative reduced cost(s)?
New SRMP
S RMP
(Short RMP)
Pricing
subproblem
New linking constraints
Primal feasibility 3 ? 7 ?
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
4 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN -D EPENDENT ROWS
(CDR) P ROBLEMS
Duals of the
missing constraints?
A subset of all columns
A subset
of all
constraints
Optimal
duals
New SRMP
S RMP
(Short RMP)
Pricing
subproblem
I
Primal feasibility 3 ? 7 ?
I
Dual feasibility 3 ? 7 ?
İbrahim Muter (CIRRELT)
Column(s) with
negative reduced cost(s)?
Simultaneous Column-and-Row Generation
New linking constraints
2012
4 / 33
Introduction
Main Motivation
M AIN M OTIVATION : C OLUMN -D EPENDENT ROWS
(CDR) P ROBLEMS
Duals of the
missing constraints?
A subset of all columns
A subset
of all
constraints
Optimal
duals
New SRMP
S RMP
(Short RMP)
Pricing
subproblem
I
Primal feasibility 3 ? 7 ?
I
Dual feasibility 3 ? 7 ?
I
Complementary slackness 3 ? 7 ?
İbrahim Muter (CIRRELT)
Column(s) with
negative reduced cost(s)?
Simultaneous Column-and-Row Generation
New linking constraints
2012
4 / 33
Introduction
Main Motivation
M AIN M OTIVATION : A N E XAMPLE
A set covering formulation, where the binary variable yk is set to 1, if column k
is selected. The parameters fk and fl are the individual contributions from
columns k and l, whereas the parameter fkl captures the cross-effects of
having columns k and l simultaneously in the solution. That is, in the objective
function we have
· · · + fk yk + fl yl + fkl yk yl + · · ·
A common linearization:
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
5 / 33
Introduction
Main Motivation
M AIN M OTIVATION : A N E XAMPLE
A set covering formulation, where the binary variable yk is set to 1, if column k
is selected. The parameters fk and fl are the individual contributions from
columns k and l, whereas the parameter fkl captures the cross-effects of
having columns k and l simultaneously in the solution. That is, in the objective
function we have
· · · + fk yk + fl yl + fkl yk yl + · · ·
A common linearization:
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
5 / 33
Introduction
Contributions
C ONTRIBUTIONS
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
6 / 33
Introduction
Contributions
C ONTRIBUTIONS
I
The problems that we refer to as CDR-problems are formulated.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
6 / 33
Introduction
Contributions
C ONTRIBUTIONS
I
The problems that we refer to as CDR-problems are formulated.
I
A set of assumptions charactarizing CDR-problems are defined.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
6 / 33
Introduction
Contributions
C ONTRIBUTIONS
I
The problems that we refer to as CDR-problems are formulated.
I
A set of assumptions charactarizing CDR-problems are defined.
I
A generic column-and-row generation algorithm for CDR-problems is
presented. The optimality of this algorithm is proved.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
6 / 33
Introduction
Contributions
C ONTRIBUTIONS
I
The problems that we refer to as CDR-problems are formulated.
I
A set of assumptions charactarizing CDR-problems are defined.
I
A generic column-and-row generation algorithm for CDR-problems is
presented. The optimality of this algorithm is proved.
I
The proposed approach is applied to the quadratic set covering, the
multi-stage cutting stock and the time-constrained routing problems.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
6 / 33
Column-Dependent-Rows Problems
Generic Mathematical Model
CDR P ROBLEMS : G ENERIC M ATHEMATICAL M ODEL
(MP)
minimize
X
ck yk +
k∈K
subject to
X
X
dn xn ,
n∈N
≥ aj ,
Ajk yk
j ∈ J,
(MP-y)
m ∈ M,
(MP-x)
k∈K
X
Bmn xn ≥bm ,
n∈N
X
k∈K
Cik yk +
X
Din xn ≥ ri ,
i ∈ I,
(MP-yx)
n∈N
yk ≥ 0, k ∈ K, xn ≥ 0, n ∈ N.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
7 / 33
Column-Dependent-Rows Problems
Generic Mathematical Model
S HORT R ESTRICTED M ASTER P ROBLEM (SRMP)
X
minimize
ck yk +
k∈K̄
X
subject to
X
dn xn ,
n∈N̄
≥ aj , j ∈ J,
Ajk yk
(SRMP-y)
k∈K̄
X
Bmn xn ≥bm , m ∈ M,
(SRMP-x)
n∈N̄
X
k∈K̄
Cik yk +
X
Din xn ≥ ri ,
i ∈ I(K̄, N̄),
(SRMP-yx)
n∈N̄
yk ≥ 0, k ∈ K̄, xn ≥ 0, n ∈ N̄,
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
8 / 33
Column-Dependent-Rows Problems
Generic Mathematical Model
S HORT R ESTRICTED M ASTER P ROBLEM (SRMP)
X
minimize
ck yk +
k∈K̄
X
subject to
X
dn xn ,
n∈N̄
≥ aj , j ∈ J,
Ajk yk
(SRMP-y)
k∈K̄
X
Bmn xn ≥bm , m ∈ M,
(SRMP-x)
n∈N̄
X
k∈K̄
Cik yk +
X
Din xn ≥ ri ,
i ∈ I(K̄, N̄),
(SRMP-yx)
n∈N̄
yk ≥ 0, k ∈ K̄, xn ≥ 0, n ∈ N̄,
I
I(K̄, N̄) ⊂ I: the set of linking constraints formed by {yk |k ∈ K̄}, and
{xn |n ∈ N̄}
I
The set of rows is dependent on the set of variables.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
8 / 33
Column-Dependent-Rows Problems
İbrahim Muter (CIRRELT)
Generic Mathematical Model
Simultaneous Column-and-Row Generation
2012
9 / 33
Column-Dependent-Rows Problems
I
Generic Mathematical Model
(SK ⊂ (K \ K̄), SN ⊂ (N \ N̄)) : the set of y and x variables to be added to
the SRMP during the column generation phase.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
9 / 33
Column-Dependent-Rows Problems
Generic Mathematical Model
I
(SK ⊂ (K \ K̄), SN ⊂ (N \ N̄)) : the set of y and x variables to be added to
the SRMP during the column generation phase.
I
∆(SK , SN ) = I(K̄ ∪ SK , N̄ ∪ SN ) \ I(K̄, N̄) : the new set of linking constraints
induced by SK and SN
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
9 / 33
Column-Dependent-Rows Problems
Generic Mathematical Model
I
(SK ⊂ (K \ K̄), SN ⊂ (N \ N̄)) : the set of y and x variables to be added to
the SRMP during the column generation phase.
I
∆(SK , SN ) = I(K̄ ∪ SK , N̄ ∪ SN ) \ I(K̄, N̄) : the new set of linking constraints
induced by SK and SN
I
The restricted master problem grows both vertically and horizontally
during column generation.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
9 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION
A SSUMPTION 1
The generation of a new set of variables {yk |k ∈ SK } prompts the generation of
a new set of variables {xn |n ∈ SN (SK )}. Furthermore, a variable
xn0 , n0 ∈ SN (SK ), does not appear in any linking constraints other than those
indexed by ∆(SK ) and introduced to the SRMP along with {yk |k ∈ SK } and
{xn |n ∈ SN (SK )}.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
10 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION
A SSUMPTION 1
The generation of a new set of variables {yk |k ∈ SK } prompts the generation of
a new set of variables {xn |n ∈ SN (SK )}. Furthermore, a variable
xn0 , n0 ∈ SN (SK ), does not appear in any linking constraints other than those
indexed by ∆(SK ) and introduced to the SRMP along with {yk |k ∈ SK } and
{xn |n ∈ SN (SK )}.
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
10 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION
A SSUMPTION 1
The generation of a new set of variables {yk |k ∈ SK } prompts the generation of
a new set of variables {xn |n ∈ SN (SK )}. Furthermore, a variable
xn0 , n0 ∈ SN (SK ), does not appear in any linking constraints other than those
indexed by ∆(SK ) and introduced to the SRMP along with {yk |k ∈ SK } and
{xn |n ∈ SN (SK )}.
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
10 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION
A SSUMPTION 1
The generation of a new set of variables {yk |k ∈ SK } prompts the generation of
a new set of variables {xn |n ∈ SN (SK )}. Furthermore, a variable
xn0 , n0 ∈ SN (SK ), does not appear in any linking constraints other than those
indexed by ∆(SK ) and introduced to the SRMP along with {yk |k ∈ SK } and
{xn |n ∈ SN (SK )}.
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
10 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION ( CONT ’ D )
A minimal variable set is a set of y−variables that triggers the generation of a
set of x−variables and the associated linking constraints in the sense of
Assumption 1.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
11 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION ( CONT ’ D )
A minimal variable set is a set of y−variables that triggers the generation of a
set of x−variables and the associated linking constraints in the sense of
Assumption 1.
A SSUMPTION 2
A linking constraint is redundant until all variables in at least one of the
minimal variable sets associated with this linking constraint are added to the
SRMP.
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
11 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION ( CONT ’ D )
A minimal variable set is a set of y−variables that triggers the generation of a
set of x−variables and the associated linking constraints in the sense of
Assumption 1.
A SSUMPTION 2
A linking constraint is redundant until all variables in at least one of the
minimal variable sets associated with this linking constraint are added to the
SRMP.
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
11 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION ( CONT ’ D )
A SSUMPTION 3
Let {yl |l ∈ SK } be a minimal variable set that generates a set of linking
constraints ∆(SK ) and a set of associated x−variables {xn |n ∈ SN (SK )}. When
the set of constraints ∆(SK ) is first introduced into the SRMP, then for each
k ∈ SK there exists a constraint i ∈ ∆(SK ) of the form
X
Cik yk +
Din xn ≥ 0,
n∈SN (SK )
where Cik > 0 and Din < 0 for all n ∈ SN (SK ).
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
12 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION ( CONT ’ D )
A SSUMPTION 3
Let {yl |l ∈ SK } be a minimal variable set that generates a set of linking
constraints ∆(SK ) and a set of associated x−variables {xn |n ∈ SN (SK )}. When
the set of constraints ∆(SK ) is first introduced into the SRMP, then for each
k ∈ SK there exists a constraint i ∈ ∆(SK ) of the form
X
Cik yk +
Din xn ≥ 0,
n∈SN (SK )
where Cik > 0 and Din < 0 for all n ∈ SN (SK ).
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
12 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION ( CONT ’ D )
A SSUMPTION 3
Let {yl |l ∈ SK } be a minimal variable set that generates a set of linking
constraints ∆(SK ) and a set of associated x−variables {xn |n ∈ SN (SK )}. When
the set of constraints ∆(SK ) is first introduced into the SRMP, then for each
k ∈ SK there exists a constraint i ∈ ∆(SK ) of the form
X
Cik yk +
Din xn ≥ 0,
n∈SN (SK )
where Cik > 0 and Din < 0 for all n ∈ SN (SK ).
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
12 / 33
Column-Dependent-Rows Problems
Assumptions
P ROBLEM C HARACTERIZATION ( CONT ’ D )
A SSUMPTION 3
Let {yl |l ∈ SK } be a minimal variable set that generates a set of linking
constraints ∆(SK ) and a set of associated x−variables {xn |n ∈ SN (SK )}. When
the set of constraints ∆(SK ) is first introduced into the SRMP, then for each
k ∈ SK there exists a constraint i ∈ ∆(SK ) of the form
X
Cik yk +
Din xn ≥ 0,
n∈SN (SK )
where Cik > 0 and Din < 0 for all n ∈ SN (SK ).
minimize
subject to
· · · + fk yk + fl yl + fkl xkl + · · ·
···
yk + yl − xkl ≤ 1, yk − xkl ≥ 0,
yl − xkl ≥ 0,
0 ≤ yk , yl , xkl ≤ 1.
···
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
12 / 33
Column-Dependent-Rows Problems
Assumptions
S OME E XAMPLE P ROBLEMS
I
Multi-Stage Cutting Stock Problem (Zak, 2002)
I
P-Median Problem (Avella et al., 2007)
I
Time-Constrained Routing Problem (Avella et al., 2006)
I
Robust Crew Pairing Problem (Muter et al., 2010)
I
Quadratic Set Covering Problem
I
Multi-Commodity Capacitated Network Design Problem (Frangioni and
Gendron, 2009)
I
Integrated Airline Recovery Problem (S. J. Maher, 2 days ago)
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
13 / 33
Column-Dependent-Rows Problems
Assumptions
M ULTI - STAGE C UTTING S TOCK (MSCS)
Stock
rolls
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
14 / 33
Column-Dependent-Rows Problems
Assumptions
M ULTI - STAGE C UTTING S TOCK (MSCS)
First
stage
cutting
Stock
rolls
İbrahim Muter (CIRRELT)
Intermediate
rolls
Simultaneous Column-and-Row Generation
2012
14 / 33
Column-Dependent-Rows Problems
Assumptions
M ULTI - STAGE C UTTING S TOCK (MSCS)
Second
stage
cutting
First
stage
cutting
Stock
rolls
İbrahim Muter (CIRRELT)
Intermediate
rolls
Simultaneous Column-and-Row Generation
Finished
rolls
2012
14 / 33
Column-Dependent-Rows Problems
Assumptions
M ULTI - STAGE C UTTING S TOCK (MSCS)
Second
stage
cutting
First
stage
cutting
Stock
rolls
Intermediate
rolls
Finished
rolls
By using the minimum number of stock rolls satisfy the demand for finished
rolls after cutting the rolls in two stages. Do not consume more intermediate
rolls than produced.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
14 / 33
Column-Dependent-Rows Problems
Assumptions
MSCS P ROBLEM
minimize
X
yk ,
k∈K
X
subject to
Bmn xn ≥bm , m ∈ M,
n∈N
X
k∈K
Cik yk +
X
Din xn ≥ 0, i ∈ I,
n∈N
yk ≥ 0, k ∈ K, xn ≥ 0, n ∈ N,
I
I
I
I
I
I: the set of intermediate rolls
M: the set of finished rolls
K: the set of cutting patterns for the first stage
N: the set of cutting patterns for the second stage
Din = −1 if a cutting pattern n for the second stage is cut from the
intermediate roll i
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
15 / 33
Column-Dependent-Rows Problems
Assumptions
MSCS P ROBLEM : SRMP
minimize
X
yk ,
k∈K̄
X
subject to
Bmn xn ≥bm , m ∈ M,
n∈N̄
X
k∈K̄
Cik yk +
X
Din xn ≥ 0, i ∈ I(K̄, N̄),
n∈N̄
yk ≥ 0, k ∈ K̄, xn ≥ 0, n ∈ N̄,
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
16 / 33
Proposed Solution Method
G ENERIC CDR M ODEL : T HE D UAL P ROBLEM
maximize
X
aj uj +
j∈J
subject to
X
X
X
bm vm +
m∈M
+
Ajk uj
ri wi ,
i∈I
X
j∈J
Cik wi ≤ck , k ∈ K,
i∈I
X
m∈M
Bmn vm +
X
Din wi ≤dn , n ∈ N,
i∈I
uj ≥ 0, j ∈ J, vm ≥ 0, m ∈ M, wi ≥ 0, i ∈ I,
I
y- and x-pricing subproblems
I
row-generating pricing subproblem
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
17 / 33
Proposed Solution Method
F LOWCHART: C OLUMN - AND -ROW G ENERATION
row−generating
Solve SRMP
Duals of the
constraints
Yes
Add columns/row(s)
to SRMP (FLAG=1)
Negatively
priced column?
Solve
row−PSP
No
No
STOP
FLAG=1?
No (FLAG=0)
START
Construct
initial SRMP
Yes
Solve SRMP
Duals of the
Yes
Solve y−PSP
constraints
Yes
Negatively
Add column(s)/row(s) priced column?
to SRMP
İbrahim Muter (CIRRELT)
FLAG=1?
No
No
(FLAG=0)
y−pricingSimultaneous Column-and-Row Generation
Negatively
priced column?
Solve x−PSP
Yes
Solve SRMP
Add column
to SRMP
(FLAG=1)
Duals of the
constraints
x−pricing
2012
18 / 33
Proposed Solution Method
y-pricing subproblem (y-PSP)
Y- PRICING SUBPROBLEM ( Y-PSP)
I
The objective is to determine a variable yk , k ∈ (K \ K̄) with a negative
reduced cost.
P
P
ζy = mink∈(K\K̄) {ck − j∈J Ajk uj − i∈I(K̄,N̄) Cik wi },
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
19 / 33
Proposed Solution Method
y-pricing subproblem (y-PSP)
Y- PRICING SUBPROBLEM ( Y-PSP)
I
The objective is to determine a variable yk , k ∈ (K \ K̄) with a negative
reduced cost.
P
P
ζy = mink∈(K\K̄) {ck − j∈J Ajk uj − i∈I(K̄,N̄) Cik wi },
I
The dual variables {uj |j ∈ J} and {wi |i ∈ I(K̄, N̄)} are obtained from the
optimal solution of the current SRMP.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
19 / 33
Proposed Solution Method
y-pricing subproblem (y-PSP)
Y- PRICING SUBPROBLEM ( Y-PSP)
I
The objective is to determine a variable yk , k ∈ (K \ K̄) with a negative
reduced cost.
P
P
ζy = mink∈(K\K̄) {ck − j∈J Ajk uj − i∈I(K̄,N̄) Cik wi },
I
The dual variables {uj |j ∈ J} and {wi |i ∈ I(K̄, N̄)} are obtained from the
optimal solution of the current SRMP.
I
If ζy is nonnegative, we move to the next subproblem.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
19 / 33
Proposed Solution Method
y-pricing subproblem (y-PSP)
Y- PRICING SUBPROBLEM ( Y-PSP)
I
The objective is to determine a variable yk , k ∈ (K \ K̄) with a negative
reduced cost.
P
P
ζy = mink∈(K\K̄) {ck − j∈J Ajk uj − i∈I(K̄,N̄) Cik wi },
I
The dual variables {uj |j ∈ J} and {wi |i ∈ I(K̄, N̄)} are obtained from the
optimal solution of the current SRMP.
I
If ζy is nonnegative, we move to the next subproblem.
I
Otherwise, there exists yk with c̄k < 0, and SRMP grows by a single
variable by setting K̄ ← K̄ ∪ {k}.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
19 / 33
Proposed Solution Method
y-pricing subproblem (y-PSP)
Y- PRICING SUBPROBLEM ( Y-PSP)
I
The objective is to determine a variable yk , k ∈ (K \ K̄) with a negative
reduced cost.
P
P
ζy = mink∈(K\K̄) {ck − j∈J Ajk uj − i∈I(K̄,N̄) Cik wi },
I
The dual variables {uj |j ∈ J} and {wi |i ∈ I(K̄, N̄)} are obtained from the
optimal solution of the current SRMP.
I
If ζy is nonnegative, we move to the next subproblem.
I
Otherwise, there exists yk with c̄k < 0, and SRMP grows by a single
variable by setting K̄ ← K̄ ∪ {k}.
I
MSCS problem: y-PSP generates a cutting pattern for the first stage that
uses only the existing intermediate rolls.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
19 / 33
Proposed Solution Method
x-pricing subproblem (x-PSP)
X - PRICING SUBPROBLEM ( X -PSP)
I
∆(∅) = ∅
ζx = minn∈NK̄
İbrahim Muter (CIRRELT)
{dn −
P
m∈M
Bmn vm −
P
i∈I(K̄,N̄)
Simultaneous Column-and-Row Generation
Din wi },
2012
20 / 33
Proposed Solution Method
x-pricing subproblem (x-PSP)
X - PRICING SUBPROBLEM ( X -PSP)
I
∆(∅) = ∅
ζx = minn∈NK̄
I
{dn −
P
m∈M
Bmn vm −
P
i∈I(K̄,N̄)
Din wi },
The dual variables {vm |m ∈ M} and {wi |i ∈ I(K̄, N̄)} are retrieved from the
optimal solution of the current SRMP.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
20 / 33
Proposed Solution Method
x-pricing subproblem (x-PSP)
X - PRICING SUBPROBLEM ( X -PSP)
I
∆(∅) = ∅
ζx = minn∈NK̄
{dn −
P
m∈M
Bmn vm −
P
i∈I(K̄,N̄)
Din wi },
I
The dual variables {vm |m ∈ M} and {wi |i ∈ I(K̄, N̄)} are retrieved from the
optimal solution of the current SRMP.
I
NK̄ is the index set of all x−variables that may be induced by the set of
variables {yk |k ∈ K̄} in the current SRMP.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
20 / 33
Proposed Solution Method
x-pricing subproblem (x-PSP)
X - PRICING SUBPROBLEM ( X -PSP)
I
∆(∅) = ∅
ζx = minn∈NK̄
{dn −
P
m∈M
Bmn vm −
P
i∈I(K̄,N̄)
Din wi },
I
The dual variables {vm |m ∈ M} and {wi |i ∈ I(K̄, N̄)} are retrieved from the
optimal solution of the current SRMP.
I
NK̄ is the index set of all x−variables that may be induced by the set of
variables {yk |k ∈ K̄} in the current SRMP.
I
If ζx < 0, N̄ ← N̄ ∪ {n},
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
20 / 33
Proposed Solution Method
x-pricing subproblem (x-PSP)
X - PRICING SUBPROBLEM ( X -PSP)
I
∆(∅) = ∅
ζx = minn∈NK̄
{dn −
P
m∈M
Bmn vm −
P
i∈I(K̄,N̄)
Din wi },
I
The dual variables {vm |m ∈ M} and {wi |i ∈ I(K̄, N̄)} are retrieved from the
optimal solution of the current SRMP.
I
NK̄ is the index set of all x−variables that may be induced by the set of
variables {yk |k ∈ K̄} in the current SRMP.
I
If ζx < 0, N̄ ← N̄ ∪ {n},
I
Otherwise, the column-and-row generation algorithm continues with the
appropriate subproblem.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
20 / 33
Proposed Solution Method
x-pricing subproblem (x-PSP)
X - PRICING SUBPROBLEM ( X -PSP)
I
∆(∅) = ∅
ζx = minn∈NK̄
{dn −
P
m∈M
Bmn vm −
P
i∈I(K̄,N̄)
Din wi },
I
The dual variables {vm |m ∈ M} and {wi |i ∈ I(K̄, N̄)} are retrieved from the
optimal solution of the current SRMP.
I
NK̄ is the index set of all x−variables that may be induced by the set of
variables {yk |k ∈ K̄} in the current SRMP.
I
If ζx < 0, N̄ ← N̄ ∪ {n},
I
Otherwise, the column-and-row generation algorithm continues with the
appropriate subproblem.
I
In the MSCS problem, the x−PSP identifies cutting patterns for the
second stage that only consume intermediate rolls that are produced by
the cutting patterns for the first stage in the current SRMP.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
20 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING PSP
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
21 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING PSP
I
Note that before invoking the row-generating PSP, no negatively priced
variables exist with respect to the current set of constraints in the SRMP.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
21 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING PSP
I
Note that before invoking the row-generating PSP, no negatively priced
variables exist with respect to the current set of constraints in the SRMP.
I
The objective of this PSP is to identify new columns that price out
favorably only after adding new linking constraints currently absent from
the SRMP.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
21 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING PSP
I
Note that before invoking the row-generating PSP, no negatively priced
variables exist with respect to the current set of constraints in the SRMP.
I
The objective of this PSP is to identify new columns that price out
favorably only after adding new linking constraints currently absent from
the SRMP.
I
The primary challenge here is to properly account for the values of the
dual variables of the missing constraints, and thus be able to determine
which linking constraints should be added to the SRMP together with a
set of variables.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
21 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING P RICING S UBPROBLEM ( CONT ’ D )
I
The optimal solution of the row-generating PSP is a family Fk of index
sets SKk .
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
22 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING P RICING S UBPROBLEM ( CONT ’ D )
I
The optimal solution of the row-generating PSP is a family Fk of index
sets SKk . For example in QSC problem, we have
Fk = {{k, l}, {k, m}},
İbrahim Muter (CIRRELT)
Σk = {k, l, m}
Simultaneous Column-and-Row Generation
2012
22 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING P RICING S UBPROBLEM ( CONT ’ D )
I
The optimal solution of the row-generating PSP is a family Fk of index
sets SKk . For example in QSC problem, we have
Fk = {{k, l}, {k, m}},
I
Σk = {k, l, m}
If the reduced cost of the variables yk corresponding to the optimal family
Fk is negative, then SRMP grows both horizontally and vertically:
SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄ ∪ Σk , N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
22 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING P RICING S UBPROBLEM ( CONT ’ D )
I
The optimal solution of the row-generating PSP is a family Fk of index
sets SKk . For example in QSC problem, we have
Fk = {{k, l}, {k, m}},
I
Σk = {k, l, m}
If the reduced cost of the variables yk corresponding to the optimal family
Fk is negative, then SRMP grows both horizontally and vertically:
SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄ ∪ Σk , N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
22 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING P RICING S UBPROBLEM ( CONT ’ D )
I
The optimal solution of the row-generating PSP is a family Fk of index
sets SKk . For example in QSC problem, we have
Fk = {{k, l}, {k, m}},
I
Σk = {k, l, m}
If the reduced cost of the variables yk corresponding to the optimal family
Fk is negative, then SRMP grows both horizontally and vertically:
SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄ ∪ Σk , N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
22 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING P RICING S UBPROBLEM ( CONT ’ D )
I
The optimal solution of the row-generating PSP is a family Fk of index
sets SKk . For example in QSC problem, we have
Fk = {{k, l}, {k, m}},
I
Σk = {k, l, m}
If the reduced cost of the variables yk corresponding to the optimal family
Fk is negative, then SRMP grows both horizontally and vertically:
SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄ ∪ Σk , N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
22 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
For any given yk , an associated Fk , and SKk ∈ Fk , we have
X
X
X
c̄k = ck −
Ajk uj −
Cik wi −
Cik wi ,
j∈J
İbrahim Muter (CIRRELT)
i∈I(K̄,N̄)
i∈∆(Σk )
Simultaneous Column-and-Row Generation
2012
23 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
For any given yk , an associated Fk , and SKk ∈ Fk , we have
X
X
X
c̄k = ck −
Ajk uj −
Cik wi −
Cik wi ,
j∈J
X
d̄n = dn −
m∈M
İbrahim Muter (CIRRELT)
i∈I(K̄,N̄)
Bmn vm −
X
i∈I(K̄,N̄)
i∈∆(Σk )
Din wi −
X
Din wi , n ∈ SN (Σk )
i∈∆(Σk )
Simultaneous Column-and-Row Generation
2012
23 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
For any given yk , an associated Fk , and SKk ∈ Fk , we have
X
X
X
c̄k = ck −
Ajk uj −
Cik wi −
Cik wi ,
j∈J
X
d̄n = dn −
i∈I(K̄,N̄)
Bmn vm −
m∈M
X
= dn −
m∈M
İbrahim Muter (CIRRELT)
X
i∈I(K̄,N̄)
Bmn vm −
X
i∈∆(Σk )
Din wi −
X
Din wi , n ∈ SN (Σk )
i∈∆(Σk )
Din wi , n ∈ SN (SKk ),
i∈∆(SKk )
Simultaneous Column-and-Row Generation
2012
23 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
For any given yk , an associated Fk , and SKk ∈ Fk , we have
X
X
X
c̄k = ck −
Ajk uj −
Cik wi −
Cik wi ,
j∈J
X
d̄n = dn −
i∈I(K̄,N̄)
Bmn vm −
m∈M
X
= dn −
m∈M
I
X
i∈I(K̄,N̄)
Bmn vm −
X
i∈∆(Σk )
Din wi −
X
Din wi , n ∈ SN (Σk )
i∈∆(Σk )
Din wi , n ∈ SN (SKk ),
i∈∆(SKk )
The values of the dual variables {uj |j ∈ J}, {vm |m ∈ M}, and
{wi |i ∈ I(K̄, N̄)} are retrieved from the optimal solution of the current
SRMP,
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
23 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
For any given yk , an associated Fk , and SKk ∈ Fk , we have
X
X
X
c̄k = ck −
Ajk uj −
Cik wi −
Cik wi ,
j∈J
X
d̄n = dn −
i∈I(K̄,N̄)
Bmn vm −
m∈M
X
= dn −
m∈M
X
i∈I(K̄,N̄)
Bmn vm −
X
i∈∆(Σk )
Din wi −
X
Din wi , n ∈ SN (Σk )
i∈∆(Σk )
Din wi , n ∈ SN (SKk ),
i∈∆(SKk )
I
The values of the dual variables {uj |j ∈ J}, {vm |m ∈ M}, and
{wi |i ∈ I(K̄, N̄)} are retrieved from the optimal solution of the current
SRMP,
I
{wi |i ∈ ∆(Σk )} are unknown.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
23 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
For any given yk , an associated Fk , and SKk ∈ Fk , we have
X
X
X
c̄k = ck −
Ajk uj −
Cik wi −
Cik wi ,
j∈J
X
d̄n = dn −
i∈I(K̄,N̄)
Bmn vm −
m∈M
X
= dn −
m∈M
X
i∈I(K̄,N̄)
Bmn vm −
X
i∈∆(Σk )
Din wi −
X
Din wi , n ∈ SN (Σk )
i∈∆(Σk )
Din wi , n ∈ SN (SKk ),
i∈∆(SKk )
I
The values of the dual variables {uj |j ∈ J}, {vm |m ∈ M}, and
{wi |i ∈ I(K̄, N̄)} are retrieved from the optimal solution of the current
SRMP,
I
{wi |i ∈ ∆(Σk )} are unknown.
I
Thinking-ahead approach: The ensuing analysis computes the optimal
values of {wi |i ∈ ∆(Σk )} without solving the SRMP explicitly under the
presence of the currently missing associated set of linking constraints
∆(Σk ).
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
23 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
I Construct an optimal basis for SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
I Construct an optimal basis for SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
1. Primal feasibility (by Assumption 2)
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
I Construct an optimal basis for SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
1. Primal feasibility (by Assumption 2)
2. Dual feasibility
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
I Construct an optimal basis for SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
1. Primal feasibility (by Assumption 2)
2. Dual feasibility
3. Complementary slackness
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
T HINKING -A HEAD : G UIDELINE
For some Fk , we take the following steps:
I SRMP(K̄, N̄, I(K̄, N̄)) ← SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
I Construct an optimal basis for SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk ))
1. Primal feasibility (by Assumption 2)
2. Dual feasibility
3. Complementary slackness
I
Calculate the correct reduced cost of yk under the condition that the
reduced costs of the variables in SRMP(K̄, N̄, I(K̄, N̄)) and Σk \{k} do not
change.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
24 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
BASIS AUGMENTATION

A1
B= 0
C1
İbrahim Muter (CIRRELT)
0
B1
D1

E1
E2 
E3
Simultaneous Column-and-Row Generation
2012
25 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
BASIS AUGMENTATION


A1
B= 0
C1
İbrahim Muter (CIRRELT)
0
B1
D1


E1

E2  → Bk = 


E3
A1
0
C1
0
C2
0
B1
D1
0
0
Simultaneous Column-and-Row Generation
E1
E2
E3
0
0
0
B2
0
D2
D3
0
0
0
0
−I






2012
25 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
BASIS AUGMENTATION


A1
B= 0
C1
I
0
B1
D1


E1

E2  → Bk = 


E3
A1
0
C1
0
C2
0
B1
D1
0
0
E1
E2
E3
0
0
0
B2
0
D2
D3
0
0
0
0
−I






New basic x-variables (⊆ SN (Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
25 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
BASIS AUGMENTATION


A1
B= 0
C1
0
B1
D1


E1

E2  → Bk = 


E3
I
New basic x-variables (⊆ SN (Σk ))
I
New basic surplus variables (⊆ ∆(Σk ))
İbrahim Muter (CIRRELT)
A1
0
C1
0
C2
0
B1
D1
0
0
Simultaneous Column-and-Row Generation
E1
E2
E3
0
0
0
B2
0
D2
D3
0
0
0
0
−I






2012
25 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
BASIS AUGMENTATION


A1
B= 0
C1
0
B1
D1


E1

E2  → Bk = 


E3
I
New basic x-variables (⊆ SN (Σk ))
I
New basic surplus variables (⊆ ∆(Σk ))
I
The dual variables should not change
İbrahim Muter (CIRRELT)
A1
0
C1
0
C2
0
B1
D1
0
0
Simultaneous Column-and-Row Generation
E1
E2
E3
0
0
0
B2
0
D2
D3
0
0
0
0
−I






2012
25 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
BASIS AUGMENTATION


A1
B= 0
C1
0
B1
D1


E1

E2  → Bk = 


E3
A1
0
C1
0
C2
0
B1
D1
0
0
E1
E2
E3
0
0
I
New basic x-variables (⊆ SN (Σk ))
I
New basic surplus variables (⊆ ∆(Σk ))
I
The dual variables should not change
I
The dual variables may be positive (∆+ (Σk ), Cik yk +
0
B2
0
D2
D3
0
0
0
0
−I
X






Din xn ≥ 0)
n∈SN (SK )
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
25 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
BASIS AUGMENTATION


A1
B= 0
C1
0
B1
D1


E1

E2  → Bk = 


E3
A1
0
C1
0
C2
0
B1
D1
0
0
E1
E2
E3
0
0
I
New basic x-variables (⊆ SN (Σk ))
I
New basic surplus variables (⊆ ∆(Σk ))
I
The dual variables should not change
I
The dual variables may be positive (∆+ (Σk ), Cik yk +
0
B2
0
D2
D3
0
0
0
0
−I
X






Din xn ≥ 0)
n∈SN (SK )
I
The dual variables should be zero (∆0 (Σk ))
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
25 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
ROW-G ENERATING PSP (T WO -L EVEL P ROBLEM )
ζyx = min
k∈(K\K̄)
αSk =maximize




ck −
X
X
Ajk uj −
j∈J
X
K
Cik wi − max 
Fk ∈Pk
i∈I(K̄,N̄)
X
k ∈F
SK
k


αSk  , where
K

(1a)
Cik wi ,
k)
i∈∆(SK
subject to
X
Din wi ≤ dn −
X
Bmn vm ,
n ∈ SN (SKk ),
(1b)
m∈M
k)
i∈∆(SK
wi = 0,
i ∈ ∆0 (SKk ),
(1c)
wi ≥ 0,
∆+ (SKk ),
(1d)
|∆(SKk )|
i∈
many linearly independent tight constraints
(1e)
among (1b)-(1d).
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
26 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
P ROPERTIES OF THE BASIS AUGMENTATION
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
27 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
P ROPERTIES OF THE BASIS AUGMENTATION
1. The optimal values of the dual variables {uj |j ∈ J}, {vm |m ∈ M}, and
{wi |i ∈ I(K̄, N̄)} are identical for SRMP(K̄, N̄, I(K̄, N̄)) and
SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
27 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
P ROPERTIES OF THE BASIS AUGMENTATION
1. The optimal values of the dual variables {uj |j ∈ J}, {vm |m ∈ M}, and
{wi |i ∈ I(K̄, N̄)} are identical for SRMP(K̄, N̄, I(K̄, N̄)) and
SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
2. The values assigned to the dual variables {wi |i ∈ ∆(Σk )} in the
row-generating PSP are optimal for SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
For i ∈ ∆s (Σk ), we have wi = 0 at optimality.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
27 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
P ROPERTIES OF THE BASIS AUGMENTATION
1. The optimal values of the dual variables {uj |j ∈ J}, {vm |m ∈ M}, and
{wi |i ∈ I(K̄, N̄)} are identical for SRMP(K̄, N̄, I(K̄, N̄)) and
SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
2. The values assigned to the dual variables {wi |i ∈ ∆(Σk )} in the
row-generating PSP are optimal for SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
For i ∈ ∆s (Σk ), we have wi = 0 at optimality.
3. The reduced costs of {yl |l ∈ K̄} and {xn |n ∈ N̄} are identical with respect
to the optimal dual solutions of SRMP(K̄, N̄, I(K̄, N̄)) and
SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
27 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
P ROPERTIES OF THE BASIS AUGMENTATION
1. The optimal values of the dual variables {uj |j ∈ J}, {vm |m ∈ M}, and
{wi |i ∈ I(K̄, N̄)} are identical for SRMP(K̄, N̄, I(K̄, N̄)) and
SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
2. The values assigned to the dual variables {wi |i ∈ ∆(Σk )} in the
row-generating PSP are optimal for SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
For i ∈ ∆s (Σk ), we have wi = 0 at optimality.
3. The reduced costs of {yl |l ∈ K̄} and {xn |n ∈ N̄} are identical with respect
to the optimal dual solutions of SRMP(K̄, N̄, I(K̄, N̄)) and
SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
4. The reduced cost c̄k computed in the row-generating PSP for any
yk , k ∈
/ K̄ and Fk ∈ Pk is equal to the reduced cost of yk with respect to the
optimal solution of SRMP(K̄, N̄ ∪ SN (Σk ), I(K̄, N̄) ∪ ∆(Σk )).
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
27 / 33
Proposed Solution Method
Row-Generating Pricing Subproblem
O PTIMALITY T HEOREM
Given an optimal basis B for SRMP(K̄, N̄, I(K̄, N̄)) and a set of associated
optimal values for the dual variables {uj |j ∈ J}, {vm |m ∈ M}, and
{wi |i ∈ I(K̄, N̄)}, the proposed column-and-row generation algorithm
terminates with an optimal solution for the master problem (MP) if ζy ≥ 0,
ζx ≥ 0, and ζyx ≥ 0 in three consecutive calls to the y−, x−, and the
row-generating PSPs, respectively.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
28 / 33
Conclusions and Future Research Directions
C ONCLUSIONS AND F UTURE R ESEARCH
I
We presented and analyzed a unified framework for large-scale linear
programs with column-dependent-rows
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
29 / 33
Conclusions and Future Research Directions
C ONCLUSIONS AND F UTURE R ESEARCH
I
We presented and analyzed a unified framework for large-scale linear
programs with column-dependent-rows
I
We identified a set of properties that characterize CDR-problems and
argued that this is indeed a large class of problems
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
29 / 33
Conclusions and Future Research Directions
C ONCLUSIONS AND F UTURE R ESEARCH
I
We presented and analyzed a unified framework for large-scale linear
programs with column-dependent-rows
I
We identified a set of properties that characterize CDR-problems and
argued that this is indeed a large class of problems
I
To the best of our knowledge, all CDR-problems studied so far in the
literature belong to this class
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
29 / 33
Conclusions and Future Research Directions
C ONCLUSIONS AND F UTURE R ESEARCH
I
We presented and analyzed a unified framework for large-scale linear
programs with column-dependent-rows
I
We identified a set of properties that characterize CDR-problems and
argued that this is indeed a large class of problems
I
To the best of our knowledge, all CDR-problems studied so far in the
literature belong to this class
I
We devised a generic methodology for solving CDR-problems to
optimality
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
29 / 33
Conclusions and Future Research Directions
C ONCLUSIONS AND F UTURE R ESEARCH ( CONT ’ D )
I
Many routing and scheduling problems with interactions between
individual routes or schedules may be cast as quadratic set covering
problems with restricted pairs and side constraints
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
30 / 33
Conclusions and Future Research Directions
C ONCLUSIONS AND F UTURE R ESEARCH ( CONT ’ D )
I
Many routing and scheduling problems with interactions between
individual routes or schedules may be cast as quadratic set covering
problems with restricted pairs and side constraints
I
We plan to develop computationally efficient implementations of our
generic methodology:
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
30 / 33
Conclusions and Future Research Directions
C ONCLUSIONS AND F UTURE R ESEARCH ( CONT ’ D )
I
Many routing and scheduling problems with interactions between
individual routes or schedules may be cast as quadratic set covering
problems with restricted pairs and side constraints
I
We plan to develop computationally efficient implementations of our
generic methodology:
I
Lagrangian relaxation and Benders decomposition
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
30 / 33
Conclusions and Future Research Directions
C ONCLUSIONS AND F UTURE R ESEARCH ( CONT ’ D )
I
Many routing and scheduling problems with interactions between
individual routes or schedules may be cast as quadratic set covering
problems with restricted pairs and side constraints
I
We plan to develop computationally efficient implementations of our
generic methodology:
I
Lagrangian relaxation and Benders decomposition
I
Embedding the column-and-row generation algorithm into branch-and-price
or branch-and-cut-and-price framework to solve large-scale integer
programming problems with column-dependent-rows.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
30 / 33
Conclusions and Future Research Directions
R ELATED R ESEARCH
I
Muter, İ., Ş. İ. Birbil, K. Bülbül. Simultaneous Column-and-Row
Generation for Large-Scale Linear Programs with
Column-Dependent-Rows, Mathematical Programming, 2012, To appear,
DOI: 10.1007/s10107-012-0561-8.
I
Muter, İ., Ş. İ. Birbil, K. Bülbül, G. Şahin. A Note on A LP-based Heuristic
for a Time-Constrained Routing Problem, European Journal of
Operational Research, 2010, 221, 2, 306-307.
I
Muter, İ., Ş. İ. Birbil, K. Bülbül, G. Şahin, D. Taş, D. Tüzün, H. Yenigün,
Solving A Robust Airline Crew Pairing Problem With Column Generation,
Computers and Operations Research, 2010, To appear, DOI:
10.1016/j.cor.2010.11.005.
İbrahim Muter (CIRRELT)
Simultaneous Column-and-Row Generation
2012
31 / 33