Journal ArticleDOI

# A Combinatorial Benders' decomposition for the lock scheduling problem

01 Feb 2015-Computers & Operations Research (Pergamon)-Vol. 54, pp 117-128

TL;DR: The results indicate that the decomposition approach significantly outperforms other exact approaches presented in the literature, in terms of solution quality and computation time.

AbstractThe Lock Scheduling Problem (LSP) is a combinatorial optimization problem that represents a real challenge for many harbours and waterway operators. The LSP consists of three strongly interconnected subproblems: scheduling lockages, assigning ships to chambers, and positioning the ships inside the chambers. These should be interpreted respectively as a scheduling, an assignment, and a packing problem. By combining the first two problems into a master problem and using the packing problem as a subproblem, a decomposition is achieved that can be solved efficiently by a Combinatorial Benders' approach. The master problem is solved first, thereby sequencing the ships into a number of lockages. Next, for each lockage, a packing subproblem is checked for feasibility, possibly returning a number of combinatorial inequalities (cuts) to the master problem. The result is an exact approach to the LSP. Experiments are conducted on a set of instances that were generated in correspondence with real world data. The results indicate that the decomposition approach significantly outperforms other exact approaches presented in the literature, in terms of solution quality and computation time.

Topics: , Optimization problem (61%), Job shop scheduling (57%), Packing problems (57%)

## Summary (4 min read)

### 1. Introduction

• It is a major hub for both inland and intercontinental cargo traffic.
• These locks entail a complex optimization problem: the vast number of ships entering and leaving the harbour every day have to be assigned to lock chambers, their exact position inside the locks need to be determined, and the lockages have to be scheduled.
• Experiments are conducted on a large number of instances based on data obtained from different Belgian locks.

### 2. Problem Outline

• The Lock Scheduling Problem consists of three interconnected sub problems: an assignment, a packing and a scheduling problem.
• Each problem comes with a large number of constraints, mainly resulting from safety and nautical regulations.
• The problem has been described in detail by Verstichel et al. (2014a); here the authors only sketch the general outline.
• The main lock scheduling specific terms used throughout this paper are elucidated in Figure 1, which shows a lock with two identical parallel chambers.

### 2.1. Scheduling, Assignment and Packing

• A lock consists of one or more chambers, which can perform lockage operations independently of each other.
• The ships must be grouped into a number of batches (i.e. lockages), where each batch contains the ships that are transferred in a single lockage operation.
• A chamber is always in one of it’s two possible states, which handle either downstream or upstream transfer.
• A first-come-firstserved (FCFS) policy is often enforced by the lock authorities.
• Therefore ship i must depart from the lock no later than ship j should ship i arrive at the lock before ship j.

### 3. Literature review

• LSP was first introduced in an inland setting by Verstichel and Vanden Berghe (2009), who presented a heuristic approach.
• Due to the complex nature of LSP, only relatively small instances were solved to optimality.
• In more recent work, e.g. Geoffrion (1972) and Hooker and Ottoson (2003), the Benders’ decomposition approach has been generalized to a broader class of problems, no longer requiring the sub problem to be linear.
• Stronger Combinatorial cuts may be obtained by identifying small subsets of variables responsible for the infeasibility of the sub problem, and expressing cuts in terms of these variables.
• Tran and Beck (2012) solve a Parallel Machine Scheduling Problem (PMSP) with machine and sequence dependent setup times through Logic Based Benders’ decomposition.

### 4. A Combinatorial Benders’ Decomposition

• Verstichel et al. (2014a) attempted to solve the LSP via a single, large, Mixed Integer Linear Programming problem.
• In addition, efficient dedicated algorithms can be employed to solve the master and sub problem, whereas there may not exist an algorithm capable of tackling the entire problem at once.
• The presented method is based on Codato and Fischetti (2006)’s original algorithm.
• Subsequently, the sub problem verifies, for each subset, whether the ships in this set can be transferred simultaneously, i.e. whether they fit to- gether inside the lock chamber.
• The LSP is solved whenever an optimal MP schedule is determined in which each subset satisfies the packing constraints of the sub problem.

### 4.1. Master problem

• The following Mixed Integer Linear Programming problem defines the master problem.
• The parameters are defined in Table 1; the variables (marked in bold) are discussed below the model.
• Other values would favor, for example, the number of lockages, maximum waiting time, etc. Constraints (2)-(4) assign ships to lockage operations.
• Constraints (9), (10) describe the actual scheduling restrictions on the lockages per chamber.
• A lockage cannot commence before all ships have arrived at the lock (Constraints (11)).

### 4.2. Sub problem

• Once the master problem has assigned the ships to a number of lockages, the feasibility of these lockages needs to be verified.
• Whenever a configuration is considered infeasible, a combinatorial cut will be generated and added to the master problem.
• The latter will be elaborated in the next section.
• Nk are located within the chamber’s dimensions.
• The remaining constraints ensure that the ships do not overlap.

### 4.3. Combinatorial Benders’ cuts

• When an infeasible sub problem is encountered, one or more combinatorial Benders’ cuts are generated and added to the master problem, effectively preventing the master problem from assigning specific ships to the same lockage.
• The strongest cuts are based on minimum infeasible subsets (MIS).
• It would require solving the sub problem from Section 4.2 for every possible subset of N ′. Section 4.4 presents approaches to computing strong cuts requiring far less computational effort.
• Combinatorial Benders’ cuts can be generated at different times in the process: Initial cuts are added to strengthen the MP before the first MPSP iteration.
• Applying initial cuts reduces the number of infeasible MP lockages generated.

### 4.4. Cut separation

• The different cut separation methods are clarified using the example from Figure 2, where the MP proposes a solution in which ships 1 through 7 are assigned to a single lockage.
• The feasible lockages for this example (under a first-come-first-served policy) are displayed on the right side of this figure.
• For the example from Figure 2, the following weak cut is generated: 7∑ i=1 fik ≤ 6, ∀k (21) Minimal infeasible subsets (MIS) can be found by applying the following constructive procedure.
• An efficient approach to identifying small infeasible subsets of ships is based on surface calculations: any set of ships having a combined surface that exceeds the total surface of the lock chamber is infeasible.
• Whenever surface calculations are used to identify infeasible subsets, it will be denoted as follows: ‘subsurf’ (subset based) and ‘surf’ (order based).

### 5. Experiments

• To assess the quality of the combinatorial Bender’s approach, a number of experiments have been conducted on instances based on real-world data originating from the Albertkanaal in Belgium (Verstichel, 2013).
• The characteristics of the locks, also known as 2. lock data.
• The ship inter arrival times have been selected from a uniform distribution between 0 and 2σ (Table 2).
• A comparison of the Benders’ procedure and the monolithic approach from Verstichel et al. (2014a) is presented.

### 5.1. FCFS single chamber lock

• The first series of experiments is performed on a single chamber lock with a first-come-first served policy for the ships.
• The x-axis of each figure displays the different instances, which are ordered, from left to right, based on (1) increasing number of ships (2) increasing inter arrival time and (3) traffic ratio (first 70/30, then 50/50).
• A slightly larger variation in the number of required iterations and cuts for equal instance size can be observed compared to the small chamber settings.
• The difference between generated cuts for the initial order cuts and the order feasibility cuts is also more significant.
• Finally, for the 48 instances consisting of 70 to 90 ships, the decomposition method solves 21 instances to optimality while for the remaining instances, feasible solutions were found.

### 5.2. No FCFS single chamber lock

• The second series of experiments is conducted on the same instances, but without the FCFS policy.
• Figures 6 (c) and (d) compare the performance of various methods.
• For the small chamber lock, initial subset cuts appear to be the best approach.
• The subship+order cut method on the other hand needed 27 cuts for the same instance, while computing only 12 seconds.
• From the above results, it is apparent that the absence of the FCFS rule has a significant impact on the computation times.

### 5.3. FCFS parallel identical chamber lock

• For the identical parallel chamber instances, the results for instances with 10 and ≥ 30 ships were omitted.
• The difference between the initial cuts is limited for a lock with two small parallel chambers ).
• The decomposition method is almost always faster for the other instances, with a total computation time of 6.5 for the monolithic approach and only 2 hours for the feasibility subset cuts.
• Applying one of the other methods (no initial cuts, subset based surface initial cuts or subset based ship placement initial cuts) results in an average of 570− 1007 cuts per instace.
• While the monolithic approach is the best choice when facing large inter arrival times.

### 5.4. FCFS multi chamber type lock

• The results for the multi chamber type lock are summarized in Figure 9.
• Here only the ≥ 30 ship instances were omitted.
• Similar to the SSC results it appears that, aside from the number of ships, the ship inter arrival time has the largest influence on the required computation time.
• Applying feasibility subset cuts combined with initial subset based surface cuts (subship+subsurf) appears to be the best way of solving LSP for this multi chamber type setting.
• This result is noteworthy, as the subship+subsurf method produces significantly more cuts than the other approaches ).

### 5.5. Heuristic sub problem approach

• The last experiment considers the effects of applying the multi-order bestfit heuristic for the ship placement problem (Verstichel et al., 2014b) to the SP.
• For the larger cases the heuristic approach matched the exact results on all but one instance, where a gap of 0.03% remained after 12 hours of computation time.
• For the single large chamber lock, the average gap for the heuristic approach on the instances with less than 60 ships was 0.10%, with a maximum of 1.19%.
• By increasing the computation time limit for these 12 instances, the authors were able to determine that in 3 cases the exact results could be matched by the heuristic approach.
• All PSC instances were solved to optimality by the heuristic decomposition method when applying initial subset cuts.

### 5.6. Summary of the experiments

• The experimental results show that the proposed Combinatorial Benders’ decomposition is very effective for the lock scheduling problem.
• This is especially the case for instances with a complex packing aspect, i.e. where several ships can be transferred in a single lockage operation, and (very) large instances.
• Indeed, for the single large chamber experiments, the authors find the largest decrease in computation time for the instances with short inter arrival times, with computation time differences of several orders of magnitude on several instances.
• Furthermore, the decomposition approach was able to produce feasible solutions, and often attest optimality, on a large number of instances that could not be tackled by the existing ‘monolithic’ approach.
• The same advantages are seen when applying a heuristic method to the sub problem.

### 6. Conclusion

• An exact Combinatorial Benders’ decomposition to the lock scheduling problem was proposed.
• The scheduling and assignment problems, whereas the packing problem is dealt with in the sub problem.
• Especially instances having ships that can be transferred simultaneously in a single lockage operation benefit from the new approach, as is shown in the experiments where the ship inter arrival times are short.
• Finally, applying a heuristic to the sub problem instead of an exact algorithm results in high quality solutions on all instances, with a maximal optimality gap of 2.93%, while the maximal time spent in the sub problem is reduced from 527 to 0.5 seconds.

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A Combinatorial Benders’ decomposition for the lock
scheduling problem
J. Verstichel
a,
, J. Kinable
a,c
, P. De Causmaecker
b
, G. Vanden Berghe
a
a
KU Leuven Department of Computer Science, CODeS, Gebroeders De Smetstraat 1,
9000 Gent, Belgium
b
KU Leuven Department of Computer Science, iMinds-ITEC, Etienne Sabbelaan 53,
8500 Kortrijk, Belgium
c
KU Leuven Faculty of Economics and Business, ORSTAT, Naamsestraat 69, 3000
Leuven, Belgium
Abstract
The Lock Scheduling Problem (LSP) is a combinatorial optimization problem
that represents a real challenge for many harbours and waterway operators.
The LSP consists of three strongly interconnected sub problems: scheduling
lockages, assigning ships to chambers, and positioning the ships inside the
chambers. These should be interpreted respectively as a scheduling, an as-
signment, and a packing problem. By combining the ﬁrst two problems into a
master problem and using the packing problem as a sub problem, a decompo-
sition is achieved that can be solved eﬃciently by a Combinatorial Benders’
approach. The master problem is solved ﬁrst, thereby sequencing the ships
into a number of lockages. Next, for each lockage, a packing sub problem is
checked for feasibility, possibly returning a number of combinatorial inequal-
ities (cuts) to the master problem. The result is an exact approach to the
LSP. Experiments are conducted on a set of instances that were generated in
correspondence with real world data. The results indicate that the decompo-
sition approach signiﬁcantly outperforms other exact approaches presented
in the literature, in terms of solution quality and computation time.
Keywords: Lock Scheduling Problem, Combinatorial Benders’
Decomposition
Email address: jannes.verstichel@cs.kuleuven.be (J. Verstichel)
Preprint submitted to Computers and Operations Research September 25, 2014

1. Introduction
The Port of Antwerp (Belgium), one of the largest harbours in Europe,
processed more than 180 MTE (Million Tonnes Equivalent) of cargo and
70000 ships in 2012, with an average of almost 200 ships a day (Port of
Antwerp, 2012). It is a major hub for both inland and intercontinental cargo
traﬃc. The harbour is situated at the river Scheldt with tidal diﬀerences
averaging ﬁve meters. In order to ensure a persistent water level within the
harbour, locks separate the docks from the main water way. These locks
entail a complex optimization problem: the vast number of ships entering
and leaving the harbour every day have to be assigned to lock chambers,
their exact position inside the locks need to be determined, and the lockages
have to be scheduled. Improving the eﬃciency of the lock operations could
reduce the expensive waiting time of ships, and make the port more attractive
for business and economy.
The Albertkanaal is an important inland waterway connecting the Port of
Antwerp with the Port of Li`ege. Over the years, numerous industrial activ-
ities have emerged on its banks, leading to over 37 MTE of cargo processed
in 2012 (nv De Scheepvaart, 2012). Six locks are used to overcome the height
diﬀerence of 56 meters between Antwerp and Li`ege. The increase of barge
traﬃc and recent periods of drought make it of paramount importance to
reduce the number of lockage operations (i.e. water usage) and the waiting
times of ships.
The present contribution is a new, fast exact approach to the lock schedul-
ing problem (LSP) based on a combinatorial Benders’ decomposition ap-
proach. The LSP is decomposed into a master and a sub problem. The
master problem (MP) ﬁrst assigns the ships to lock chambers, after which it
attempts to schedule lockages. The sub problem (SP) takes care of position-
ing the ships inside the lock chambers. Whenever the sub problem identiﬁes
an infeasible lockage, i.e. a set of ships that cannot be transferred simulta-
neously due to the chamber’s capacity or safety constraints, combinatorial
inequalities (cuts) are generated and added to the master problem. The mas-
ter problem and sub problem are solved iteratively, until a provable optimal
schedule is obtained.
The main focus of this paper is on the decomposition approach and its
application to LSP, thereby omitting detailed discussions on the sub problems
as they exhibit a large number of application speciﬁc constraints. Section 2
describes the LSP. Section 3 provides a literature review. Section 4 presents
2

the Benders’ decomposition approach, deﬁning the master problem and the
sub problem in detail, as well as their interaction. Particular attention goes
to the generation of feasibility cuts for the master problem, as they largely
determine the eﬃciency of the algorithm. Experiments are conducted on
a large number of instances based on data obtained from diﬀerent Belgian
locks. The results are presented in Section 5. Section 6 oﬀers the conclusions.
2. Problem Outline
The Lock Scheduling Problem consists of three interconnected sub prob-
lems: an assignment, a packing and a scheduling problem. Each problem
comes with a large number of constraints, mainly resulting from safety and
nautical regulations. The problem has been described in detail by Verstichel
et al. (2014a); here we only sketch the general outline. The main lock schedul-
ing speciﬁc terms used throughout this paper are elucidated in Figure 1,
which shows a lock with two identical parallel chambers.
High levelLow level
Ship 4
Ship 1
Ship 2
Chamber 1
Ship 3
x
y
Chamber 2
Width
Length
Ship 1 @ (75,100)
Lockage 1
Ship 2 @ (0,50)
Ship 3 @ (x3,y3)
Lockage 2 Lockage operation 1
Lockage 1 @ 9h00
on Chamber 2
Lockage operation 2
Lockage 2 @ 9h35
on Chamber 1
Figure 1: An example of a lock with two parallel chambers (|T | = 1, |U
1
| = 2), four ships
travelling through the lock (|N
1
| = 3, |N
2
| = 1) and important lock scheduling speciﬁc
terms.
3

2.1. Scheduling, Assignment and Packing
A lock consists of one or more chambers, which can perform lockage
operations independently of each other. Each chamber is of a speciﬁc type
t T , deﬁning the chamber’s dimensions, transfer speed, etc. The set of
chambers of the same type is denoted by U
t
.
A number of ships N need to traverse the lock, either in the upstream, or in
the downstream direction. Upstream (resp. downstream) ships are denoted
as N
1
(resp. N
2
), N
1
N
2
= , N
1
N
2
= N. For each ship i N, an
arrival time r
i
is known, as well as the dimensions of the ship. The ships
must be grouped into a number of batches (i.e. lockages), where each batch
contains the ships that are transferred in a single lockage operation. The set
of all lockage operations is denoted by M, thereby distinguishing upstream
M
1
and M
2
downstream lockages (M = M
1
M
2
). Each lockage operation
k M needs to be assigned to a physical chamber u U
t
that will execute
the lockage. We therefore distinguish between lockages M
t
M of a speciﬁc
type t T , i.e. lockage operations that can be performed on a chamber of
type t T .
All lockages M
t
, t T need to be distributed over the available chambers
U
t
, while adhering to a strict schedule. A lockage k M cannot commence
before the last ship assigned to the lockage has arrived. The duration of the
lockage depends both on the processing times ps
i
of the ships i N assigned
to the lockage, plus a constant time pc
t
depending on the type t T of
chamber that performs the lockage.
A chamber is always in one of it’s two possible states, which handle either
downstream or upstream transfer. Each transfer switches the lock’s state.
Consequently, two consecutive upstream (or downstream) lockage operations
on the same chamber require an empty lockage to be scheduled in between
to switch the chamber’s state. Formally, given two consecutive lockages,
k, l M
t
, scheduled on a chamber of type t T , a transition time s
lk
is
needed to put the chamber in the correct state, i.e. s
lk
= pc
t
if k, l are both
upstream (or downstream) lockages, s
lk
= 0 otherwise. A ﬁrst-come-ﬁrst-
served (FCFS) policy is often enforced by the lock authorities. Therefore
ship i must depart from the lock no later than ship j should ship i arrive at
the lock before ship j. It is assumed that no two ships ever arrive at the lock
at exactly the same time.
Assigning ships to a speciﬁc lockage operation requires evaluation of a
number of constraints. Obviously, the ships assigned to a single lockage
k M
t
, may not surpass the capacity of a chamber of type t T . In
4

addition, the exact location of a ship inside the chamber u U
t
has to
be determined, while complying with a number of safety restrictions. In
short, verifying whether a set of ships can be assigned to the same lockage
operation, amounts to solving a complex rectangle packing problem, where
each rectangle represents a ship (Verstichel et al., 2014b).
LSP corresponds to a traditional machine scheduling problem with se-
quence dependent setup times: a set of tasks (ships) are grouped into jobs
(lockages) which need to be assigned to machines (lock chambers).
An overview of the parameters of the LSP is provided in Table 1.
Parameters:
N, N
1
, N
2
N = N
1
N
2
is the set of ships, subdivided in upstream
ships N
1
and downstream ships N
2
.
T Set of diﬀerent chamber types.
M M = M
1
M
2
is the set of lockages, thereby distinguish-
ing between upstream M
1
and downstream M
2
lockages.
M
t
M
t
= M
1
t
M
2
t
is the set of lockages suitable for cham-
bers of type t T , again distinguishing between resp.
upstream and downstream lockages. Note that M
t
is an
ordered set, i.e. M
t
= {1, 2, . . . , m
1
t
, m
1
t
+1, . . . , m
1
t
+m
2
t
},
where m
i
t
, i = 1, 2, are bounds on the number of
upstream resp. downstream lockages for chamber type
t T .
U
t
Set of chambers of type t T .
pc
t
, ps
i
The minimal processing time of a chamber of type t T ,
and the processing time of ship i N.
r
i
Time at which ship i N arrives at the lock.
s
lk
Transition time required between two consecutive lock-
ages k, l M
t
, t T performed on the same chamber.
s
lk
= pc
t
if k, l are both upstream (or downstream) lock-
ages, s
lk
= 0 otherwise.
W
t
, L
t
: Width and length of a chamber of type t T (integer)
w
i
, l
i
: Width and length of ship i N (integer)
C
max
: Suﬃciently large big-m parameter
Table 1: Parameters used throughout the paper.
5

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Journal ArticleDOI
TL;DR: The aim of this paper is to generalize the linear programming dual used in the classical method to an ``inference dual'' that takes the form of a logical deduction that yields Benders cuts.
Abstract: Benders decomposition uses a strategy of ``learning from one's mistakes.'' The aim of this paper is to extend this strategy to a much larger class of problems. The key is to generalize the linear programming dual used in the classical method to an ``inference dual.'' Solution of the inference dual takes the form of a logical deduction that yields Benders cuts. The dual is therefore very different from other generalized duals that have been proposed. The approach is illustrated by working out the details for propositional satisfiability and 0-1 programming problems. Computational tests are carried out for the latter, but the most promising contribution of logic-based Benders may be to provide a framework for combining optimization and constraint programming methods.

409 citations

### "A Combinatorial Benders' decomposit..." refers background in this paper

• ...Hooker and Ottoson (2003) extends the application of Benders’ decomposition even further through the introduction of inference duals and a logic-based Benders’ decomposition method....

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Journal ArticleDOI
TL;DR: Computational results on two specific classes of hard-to-solve MIPs indicate that the new method produces a reformulation which can be solved some orders of magnitude faster than the original MIP model.
Abstract: Mixed-integer programs (MIPs) involving logical implications modeled through big-M coefficients are notoriously among the hardest to solve. In this paper, we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the big-M coefficients. Our solution scheme defines a master integer linear problem (ILP) with no continuous variables, which contains combinatorial information on the feasible integer variable combinations that can be “distilled” from the original MIP model. The master solutions are sent to a slave linear program (LP), which validates them and possibly returns combinatorial inequalities to be added to the current master ILP. The inequalities are associated to minimal (or irreducible) infeasible subsystems of a certain linear system, and can be separated efficiently in case the master solution is integer. The overall solution mechanism closely resembles the Benders' one, but the cuts we produce are purely co...

297 citations

### "A Combinatorial Benders' decomposit..." refers background or methods in this paper

• ...The main differences are that we work with a integer programming sub problem instead of a linear one, and that we apply a constructive algorithm for determining minimal infeasible subsets, whereas Codato and Fischetti (2006) determined MIS through an LP....

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• ...Codato and Fischetti (2006) introduce a similar solution approach for mixed-integer programming (MIP) problems involving logical implications (big M constraints), but instead of generating cuts through solving the dual of the linear programming sub problem they rely on the generation of minimal…...

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• ...Finally, Section 6 offers the conclusions....

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• ...The presented method is very similar to that of Codato and Fischetti (2006)....

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Journal ArticleDOI
TL;DR: A hybrid IP/CP algorithm for designing a double round robin schedule with a minimal number of breaks is presented and is capable of solving a number of previously unsolved benchmark problems for the Traveling Tournament Problem with constant distances.
Abstract: This paper presents a hybrid IP/CP algorithm for designing a double round robin schedule with a minimal number of breaks. Both mirrored and non-mirrored schedules with and without place constraints are considered. The algorithm uses Benders cuts to obtain feasible home-away pattern sets in few iterations and this approach leads to significant reductions in computation time for hard instances. Furthermore, the algorithm is capable of solving a number of previously unsolved benchmark problems for the Traveling Tournament Problem with constant distances.

75 citations

### "A Combinatorial Benders' decomposit..." refers background in this paper

• ...Hooker (2007) and Rasmussen and Trick (2007) combine mixedinteger linear programming and constraint programming in hybrid Benders’ decomposition frameworks and substantially improve on the state of the art in their respective application domains....

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##### Frequently Asked Questions (2)
###### Q1. What contributions have the authors mentioned in the paper "A combinatorial benders’ decomposition for the lock scheduling problem" ?

In this paper, the authors proposed a new exact approach to the lock scheduling problem ( LSP ) based on a combinatorial Benders ' decomposition approach.

Future work may be aimed at improving the MP ’ s procedure, which is currently the main bottleneck of the decomposition approach.