# Modeling and solving the Aircraft Landing Problem // 12 Oct 2020

In this post, we present the model and the results of LocalSolver on the Aircraft Landing Problem. This problem is described in the article "Scheduling aircraft landings - the static case" by J. E. Beasley, M. Krishnamoorthy, Y. M. Sharaiha, and D. Abramson (2000), Transportation Science 34(2), pp. 180-197. The test instances come from the OR-Library maintained by J. E. Beasley.

In the Aircraft Landing Problem, landing times of a set of planes have to be scheduled. Each plane can land in a predetermined time window around a target time. The objective is to minimize the “cost penalty” due to landings before or after target times. A separation time has to be respected between two successive planes. This separation time depends on the types of planes. This diagram presents an example of a solution to the problem with 3 planes.

Diagram of a solution for an aircraft landing instance

In this LocalSolver model, here implemented in Python, the landing order is represented as a list variable. The i-th element of the list corresponds to the index of the i-th plane to land. To ensure that all planes are scheduled, the size of the list variable is constrained to be equal to the number of planes.

``````# A list variable: landingOrder[i] is the index of the ith plane to land
landing_order = model.list(nb_planes)

# All planes must be scheduled
model.constraint(model.count(landing_order) == nb_planes)``````

Another decision variable is introduced: the preferred landing time for each plane. A plane should land after the target time only if it has to wait after the landing of the preceding plane. Thus, the preferred landing time of a plane is between the earliest landing time and the target time.

``````# Int variable: preferred landing time for each plane
preferred_time = [model.int(earliest_time[p], target_time[p]) for p in range(nb_planes)]
preferred_time_array = model.array(preferred_time)``````

The landing time of each plane is defined by a recursive function as the maximum between the preferred landing time and the first time the plane can land, respecting the separation time from the previous plane.

``````# Landing time for each plane
landing_time_selector = model.lambda_function(lambda p, prev:
model.max(preferred_time_array[landing_order[p]],
get_min_landing_time(p, prev, model, separation_time_array,
landing_order)))

landing_time = model.array(model.range(0, nb_planes), landing_time_selector)``````

Landing times must respect the separation time with every previous plane.

``````for p in range(1, nb_planes):
last_separation_end = [landing_time[previous_plane] +
model.at(separation_time_array, landing_order[previous_plane], landing_order[p])
for previous_plane in range(p)]
model.constraint(landing_time[p] >= model.max(last_separation_end))``````

The objective is to minimize the penalty cost due to the earliness or the lateness of each plane. To compute this cost for each plane, a ternary condition is used to select either the earliness cost or the lateness cost depending on the difference between the landing time and the target time.

Finally, we initialize the list with the planes sorted by target times.

``````# Initialize landing_order
list_to_init = landing_order.get_value()
list_to_init.clear()
sorted_planes_by_target = sorted(range(len(target_time)), key = lambda k: target_time[k])
for p in sorted_planes_by_target :

Optimal solutions are known for the first 8 instances. For instances 9-13, the gap in % is computed from the best solutions reported in the literature. For the first 8 instances, LocalSolver finds the optimal solution within a few seconds. For larger instances, from 100 to 500 planes, the gap in % to the best-known solution for different time limits are presented in the figure below.

Instances #Planes Best known
solutions
1 min 5 min 30 min 60 min
1 10 700 0.0 0.0 0.0 0.0
2 15 1,480 0.0 0.0 0.0 0.0
3 20 820 0.0 0.0 0.0 0.0
4 20 2,520 0.0 0.0 0.0 0.0
5 20 3,100 0.0 0.0 0.0 0.0
6 30 24,442 0.0 0.0 0.0 0.0
7 44 1,550 0.0 0.0 0.0 0.0
8 50 1,950 0.0 0.0 0.0 0.0
9 100 5,611 1.1 0.0 0.0 0.0
10 150 12,292 4.2 3.0 0.8 0.7
11 200 12,418 6.2 3.1 3.0 0.8
12 250 16152 5.7 1.7 0.8 0.0
13 500 37,268 25.4 8.3 1.7 1.6

LocalSolver's solutions on the Aircraft Landing Problem

This last figure presents the results obtained on instance 9 with 100 planes.

LocalSolver's solution on the instance 9 with 100 planes

You can find the complete model of the Aircraft Landing Problem in Python, Java, C#, C++, and LSP in our Example Tour:

We are at your disposal to accompany you in the discovery of LocalSolver. Don't hesitate to contact us for any further information or support.

# Other blog posts

## 2021

### LocalSolver Cloud // 21 Jul 2021

The team is glad to introduce LocalSolver Cloud, the easiest and cheapest way on the market to leverage the power of cloud computing for your optimization projects.

### New release: LocalSolver 10.5 // 22 Jun 2021

Léa, Optimization Scientist, presents the new release 10.5, which comes with many new features and performance improvements. Register and have a try for free now!

### LocalSolver at VivaTech 2021 // 31 May 2021

So glad to be invited by our client Air Liquide to VivaTech 2021, Europe’s biggest startup and tech event. Come to discover how LocalSolver solves large-scale Inventory Routing Problems in minutes, making bulk gas transportation both greener and cheaper.

### Welcome to our new clients // 20 May 2021

LocalSolver is the premier solver for supply chain management problems. Several major companies have recently switched to LocalSolver to solve their most challenging optimization problems. Find out which companies trust us.

### Modeling and solving Curve Fitting problems // 26 Apr 2021

LocalSolver solves combinatorial but also continuous optimization problems. Here we show how Curve Fitting problems can be easily modeled and optimally solved, in seconds, using LocalSolver.

### Two new big features for LSP lovers // 08 Apr 2021

LSP is an innovative modeling language coming on top of the LocalSolver and its APIs. With LocalSolver 10.0, we released two major features enhancing the LSP experience: the JSON module and the Modeler API. Find out how to use them.

### Welcome to our new clients // 11 Mar 2021

LocalSolver is the premier solver for supply chain management problems. Several major companies have recently switched to LocalSolver to solve their most challenging optimization problems. Find out which companies trust us.

### Solving constrained black-box optimization problems // 01 Mar 2021

Surrogate modeling is now fully integrated into LocalSolver, making it much more efficient and easier to use. In this post, look at the performance obtained on a simulation optimization problem coming from Revenue Management.

### Lower bounds for Vehicle Routing Problems // 11 Feb 2021

LocalSolver keeps on improving its capabilities to solve Vehicle Routing Problems (VRP). LocalSolver now delivers tight lower bounds quickly for large-scale vehicle routing problems.

### New release: LocalSolver 10.0 // 18 Jan 2021

Léa, Optimization Scientist, presents the new release 10.0, which comes with many new features and performance improvements. Register and have a try for free now!

### Welcome to our new clients // 11 Jan 2021

LocalSolver is the premier solver for supply chain management problems. Several major companies have recently switched to LocalSolver to solve their most challenging optimization problems. Find out which companies trust us.

## 2020

### LocalSolver keeps on breaking records! // 22 Nov 2020

Here is another piece of evidence that LocalSolver is the best solver to tackle scheduling problems. Using a straightforward modeling approach, LocalSolver improves the best-known solutions for 257 open instances of the Assembly Line Balancing Problem.

### Interested in doing an internship at LocalSolver? // 06 Nov 2020

Are you passionate about Operations Research and Mathematical Optimization? Have a look at our internship offers for the year 2021.

### Modeling and solving the Aircraft Landing Problem // 12 Oct 2020

LocalSolver is the solver of choice to tackle large-scale scheduling problems. As an example, we show here how the Aircraft Landing Problem can be easily modeled and efficiently solved using LocalSolver.

### Global Optimization with LocalSolver // 22 Sep 2020

We are pleased to announce the defense of the doctoral thesis of Simon Boulmier, optimization engineer at LocalSolver. This presentation will conclude our doctoral student's 3 years of research on the theme of Global Optimization with LocalSolver.

### Broken records for the Movie Shoot Scheduling Problem // 09 Sep 2020

We present a new modeling example illustrating the strength of LocalSolver’s external functions to concisely model and efficiently solve complex optimization problems.

### Solution Repair by Inequality Network Propagation in LocalSolver // 31 Aug 2020

Léa, optimization expert at LocalSolver, will give a talk on Solution Repair by Inequality Network Propagation in LocalSolver at the Sixteenth International Conference on Parallel Problem Solving from Nature (PPSN 2020).

### LocalSolver at SMAI-MODE 2020 // 15 Jul 2020

LocalSolver will participate in the 2020 workshop on Mathematics for Optimization and Decision (MODE), part of the French Society for Applied and Industrial Mathematics (SMAI), September 7-9, 2020.

### LocalSolver modeling guide for Vehicle Routing Problems // 15 Jun 2020

LocalSolver is a generic optimization solver with a simple and powerful formalism. Its genericity offers unlimited possibilities to model and solve real-life vehicle routing problems.

### Huge vehicle routing instances // 18 May 2020

We analyze the performance of LocalSlover on huge instances of the Capacitated Vehicle Routing Problem (CVRP). The instances come from a paper by Arnold, Gendreau, Sörensen published in July 2019.

### LocalSolver extension for Visual Studio Code // 16 Apr 2020

Need to quickly prototype? Use the LocalSolver Programming language (LSP) through Visual Studio Code. LSP makes the modeling of your problem easier and faster.

### Non-convex quadratic models // 18 Mar 2020

We present results obtained on non-convex quadratic models recently discussed on Erwin Kalvelagen's blog. LocalSolver is shown to be much easier to use, but also much better and faster, than Cplex and Gurobi.

### New release: LocalSolver 9.5 // 02 Mar 2020

LocalSolver 9.5 comes with big novelties. Have a look to the new features and performance improvements it offers. And try it for free now!

## 2019

### LocalSolver sponsors Dataquitaine 2020 // 09 Dec 2019

LocalSolver sponsors Dataquitaine, February 13, in Bordeaux, about the practice and impact of optimization and operations research in business and industry.

### LocalSolver at ORBEL34 // 14 Nov 2019

LocalSolver is glad to sponsor ORBEL34, the 34th annual conference of Belgian Operational Research Society at UCL-CORE, in January 30-31.

LocalSolver will sponsor the conference ROADEF 2020, February 19-21. Come to meet us on our booth, to discover LocalSolver 9.5 new features and applications.

### Interested in doing an internship with us? // 01 Sep 2019

Are you passionate about Operations Research and Computer Science? Have a look to our internship offers for the year 2020. Then join us!

### LocalSolver opens 3 positions as Optimization Engineer // 01 Sep 2019

Passionate about optimization? Join a small, fast-growing tech company which is changing the game in the world of optimization.

### We are hiring our Marketing & Communication Manager // 01 Sep 2019

Do you love communicating about science and technology? Then join a highly-skilled, fast-growing team to push forward the most innovative software project in the field.

### LocalSolver at OR2019 in Dresden // 23 Aug 2019

LocalSolver team will be at OR2019 in Dresden, June 23-26. Come to meet the team and discover the next 9.0 version.

### New release: LocalSolver 9.0 // 01 Aug 2019

This new release offers better solutions and better bounds for many optimization problems, especially combinatorial problems (packing, routing, scheduling) as well as nonlinear continuous problems. Try it for free now!

### LocalSolver at EURO 2019 in Dublin // 22 Jun 2019

LocalSolver team will be at EURO 2019 in Dublin, June 23-26. Come to meet the team and discover the next 9.0 version.

### LocalSolver at ROADEF 2019 // 18 Feb 2019

The team will be at the conference ROADEF 2019, February 19-21. Come to meet us on our booth, to discover LocalSolver 8.5 new features and applications.

### New release: LocalSolver 8.5 // 29 Jan 2019

This new release offers better bounds for bool, int, and float-based models and better solutions for set and list-based models. Try it for free now!

## 2018

### LocalSolver will be at OR2018 in Brussels // 03 Sep 2018

The team will be at the conference OR2018 in Brussels, September 12-14. Come to meet us on our booth, to discover LocalSolver 8.0 new features and applications.

### New release: LocalSolver 8.0 // 02 Aug 2018

We are pleased to announce the release of LocalSolver 8.0, which introduces set variables in addition to list variables. Try it for free now!

### LocalSolver at ISMP 2018 in Bordeaux // 02 Jul 2018

LocalSolver team will be at ISMP 2018 in Bordeaux, July 2-6. Come to our exhibit booth to meet the team and discover the next 8.0 version.

### LocalSolver is hiring a Numerical Optimization Engineer // 23 Jun 2018

Passionate about numerical optimization? Join a small, fast-growing tech company which is changing the game in the world of optimization.

### LocalSolver is hiring a Full-Stack Developer // 30 May 2018

Passionate about computer programming? Join a small, fast-growing tech company which is changing the game in the world of optimization.

### LocalSolver finalist at Les PROS de la RO // 28 Mar 2018

We are glad to be selected as finalist of the French Operations Research Society competition for the best OR project in industry.

## 2017

### New release: LocalSolver 7.5 // 20 Nov 2017

We are pleased to announce the release of LocalSolver 7.5, which comes with big novelties. Try it for free now!

### Interested in doing an internship at LocalSolver? // 04 Oct 2017

Are you passionate about Operations Research and Mathematical Optimization? Have a look to our internship offers for the year 2018. Then join us!

### LocalSolver sponsors OR2017 in Berlin // 31 Aug 2017

The team will be at the conference OR2017 in Berlin, September 6-8. Come to meet us on our booth, to discover LocalSolver 7.0 new features and applications.

### New release: LocalSolver 7.0 // 02 May 2017

LocalSolver 7.0 offers new features and performance improvements for a bunch of combinatorial and numerical problems. Try it for free now!

## 2016

### Welcome to our new clients // 31 Oct 2016

Several organizations have recently chosen LocalSolver to solve their most challenging optimization problems. Discover companies and institutions who trust us.

### New release: LocalSolver 6.5 // 29 Jul 2016

LocalSolver 6.5 offers big performance improvements for routing & scheduling problems, as well as for almost-linear problems. Check it out for free now!

### Solving a problem with 8 million boolean decisions // 15 Jun 2016

This is the number of binary decisions of the supply chain optimization problem solved in minutes by Pasco Shikishima and Future Architect, using LocalSolver.

### We search for a business developer // 16 Mar 2016

Interested in selling mathematical software? Interested in joining a fast-growing tech company? Have a look to this open position at Innovation 24 & LocalSolver.

### Welcome to our new clients // 01 Mar 2016

Several organizations have recently chosen LocalSolver to solve their most challenging optimization problems. Discover companies and institutions who trust us.

### New release: LocalSolver 6.0 // 09 Feb 2016

LocalSolver 6.0 offers new unique, powerful features: solve routing & scheduling problems at hand using list variables, plug external functions easily to your optimization models, analyze efficiently why your model is infeasible, ...

LocalSolver 6.0 will be unveiled at ROADEF 2016, the annual congress of the French Operations Research Society, in Compiègne, 10-12th February 2016.

### LocalSolver sponsors ORBEL // 25 Jan 2016

LocalSolver is glad to sponsor ORBEL30, the 30th annual conference of Belgian Operationnal Research Society at UCL-CORE, January 28-29, 2016.

## 2015

### FICO Kaggle Challenge: Santa's Stolen Sleigh // 10 Dec 2015

The analytics company FICO has launched an optimization challenge on Kaggle: Santa's Stolen Sleigh. Julien Darlay, from LocalSolver R&D, proposes a simple optimization approach based on LocalSolver to solve this problem nearly optimally.

### LocalSolver at PGMO DAYS 2015 // 25 Sep 2015

LocalSolver will be at PGMO DAYS 2015. Discover how LocalSolver solves a real-life network optimization problem with 14,000 nodes and 180,000 edges in minutes.

### LocalSolver sponsors OR2015 in Vienna // 16 Jul 2015

LocalSolver sponsors OR2015 in Vienna. Come on our booth to meet the us and discover the big novelties released in LocalSolver 5.5 version!

### LocalSolver 5.5 is released // 11 Jul 2015

Have a look to this new version which offers new modeling features, as well as performance improvements for a bunch of hard problems.

### Welcome to our new clients // 03 Jul 2015

Several organizations have newly chosen LocalSolver to solve their most challenging optimization problems. Discover the corporations and institutions who trust us.

### LocalSolver sponsors EURO 2015 // 30 Jun 2015

LocalSolver sponsors the biggest OR event in Europe: EURO 2015. Come on our booth to meet the us and discover the big novelties coming in LocalSolver 5.5 version!

### LocalSolver at MIC 2015 // 15 Apr 2015

Julien Darlay will give a plenary talk at 11th Metaheuristics International Conference (MIC 2015) in Agadir, Morocco, from the 7 to 10th June 2015.

### Applications of Optimization 2015 // 31 Mar 2015

Frédéric Gardi will give a talk at the Applications of Optimization 2015 workshop organized by the Danish Operations Research Society (DORS).

### New release: LocalSolver 5.0 // 19 Jan 2015

LocalSolver 5.0 is out. Have a look to this new version which offers new modeling features, as well as performance improvements for a bunch of hard problems.

### Je suis Charlie // 11 Jan 2015

We are shocked and depressed by the terrible events that occurred in recent days in France. We have a thought for the victims and their families. #JeSuisCharlie

## 2014

### Welcome to our new clients // 20 Dec 2014

Several organizations have newly chosen LocalSolver to solve their most challenging optimization problems. Discover the corporations and institutions who trust us.

### LocalSolver hires: welcome to Clément! // 10 Nov 2014

LocalSolver hires: our team is pleased to welcome Clément Pajean, graduated from ENSTA ParisTech, one of the French "Grandes Ecoles".

### Plugging LocalSolver to the R language // 12 Aug 2014

Our Polish partner and reseller WLOG Solutions has developed the localsolver package for the R language.

### New release: LocalSolver 4.5 // 21 Jul 2014

We invite all our users to install LocalSolver 4.5. This new version offers new modeling features, as well as performance and numerical precision improvements.

### LocalSolver sponsors OR 2014 // 25 Jun 2014

LocalSolver sponsors OR 2014 in Aachen in Germany, from 2 to 5th September 2014. Come to our booth to discover LocalSolver and meet the team!

### LocalSolver at EUROPT 2014 // 07 May 2014

LocalSolver will be presented at the 12th EUROPT Workshop on Advances in Continuous Optimization, July 10-12 2014, in Perpignan, France.

### LocalSolver welcomes new clients // 23 Apr 2014

Several organizations have recently chosen LocalSolver as premier mathematical optimization solver: discover them!

### LocalSolver at SMAI-MODE 2014 // 30 Mar 2014

The team was invited to present LocalSolver at the 2014 workshop on Mathematics for Optimization and Decision (MODE), part of the French Society for Applied and Industrial Mathematics (SMAI), March 26-28th 2014 in Rennes, France.

### LocalSolver sponsors INFORMS 2014 // 14 Mar 2014

The LocalSolver team will have a booth and talk at INFORMS 2014 in Boston, Massachusetts. Come to meet us.

### LocalSolver at Paris Machine Learning Meetup // 12 Mar 2014

LocalSolver team was recently invited to the Paris Machine Learning Meetup organized by Igor Caron, to talk about large-scale optimization related to machine learning.

### Thierry Benoist's Habilitation // 06 Mar 2014

Thierry Benoist, member of the LocalSolver team, will defend his Habilitation Thesis on March 11th 2014 at Ecole des Mines de Nantes (France).

### LocalSolver sponsors JFPC-JIAF 2014 // 01 Mar 2014

LocalSolver sponsors the French Meeting on Constraint Programming and Articficial Intelligence, from 11th to 13th June 2014 in Angers, France.

### LocalSolver at ROADEF 2014 // 27 Jan 2014

LocalSolver will be at ROADEF 2014, the annual congress of the French Operations Research Society, in Bordeaux from 26th to 28th February. Come to meet the team!

### LocalSolver at Workshop ORO 2013 // 03 Jan 2014

Frédéric Gardi was invited as plenary speaker to the ORO Workshop in Nantes. He spoke about his research line on "how industrializing local search-based OR software?" which leads to the genesis of LocalSolver.

## 2013

### Merry Christmas: LocalSolver 4.0 is released! // 25 Dec 2013

We wish a Merry Christmas to all our clients and users: LocalSolver 4.0 is out! This new version allows to tackle large-scale mixed-variable non-convex optimization problems.

### Frédéric Gardi's Habilitation // 25 Nov 2013

Frédéric Gardi, member of the LocalSolver team, has defended his Habilitation Thesis on November 25th 2013 at Université Pierre et Marie Curie (Paris 6, Jussieu).

### Innovation 24, the parent company of LocalSolver // 17 Oct 2013

Discover Innovation 24, the parent company of LocalSolver, also offering expert services in business analytics and operations research.

### LocalSolver wins against MILP, MIQP, and CP solvers on QAPLIB // 01 Oct 2013

While MILP, MIQP or CP solvers are out of scope, LocalSolver reaches state-of-the-art results within 5 minutes on Quadratic Assignment Problem.

### Welcome to our new clients // 24 Sep 2013

We welcome our new clients: Hitachi, MediaTransports (Publicis), NIES, Nanjin University.

### LocalSolver tutorial in IFORS Newsletter // 23 Sep 2013

IFORS is the International Federation of Operational Research Societies. A tutorial on LocalSolver appears in the last IFORS Newsletter.

### Welcome to our new partners // 19 Sep 2013

We are glad to welcome new LocalSolver partners.

### LocalSolver at the Summer School "Francesco Turco" 2013 // 10 Sep 2013

Our Italian partner LAC is sponsoring of Summer School "Francesco Turco" 2013. LocalSolver will be presented during the event.

### LocalSolver at OR55 in Exeter (UK) // 13 Aug 2013

LocalSolver is sponsoring OR55 in Exeter (UK), 3-5th September 2013. Come to our booth to meet the team!

### LocalSolver at OR 2013 in Rotterdam // 30 Jul 2013

LocalSolver is sponsoring OR 2013 in Rotterdam, 3-6th September 2013. Come to our booth to meet the team!

### LocalSolver 3.1 is out // 28 Jun 2013

LocalSolver 3.1 is now available. Enjoy it!

### LocalSolver at EURO-INFORMS 2013 // 15 Jun 2013

LocalSolver sponsors EURO-INFORMS 2013 in Rome. Don't miss the presentations made by the team!

### Speed dating with LocalSolver // 06 Jun 2013

A nice post about LocalSolver by the OR blogger Marc-André Carle.

### LocalSolver at ECCO 2013 // 30 Apr 2013

Don't miss the presentation of LocalSolver at ECCO 2013 in Paris.

### Welcome to our new clients // 02 Apr 2013

We welcome our new clients: Air Liquide, Armée de Terre, Fujitsu.

## 2012

### Customers & Partners // 14 Nov 2012

Welcome to our new customers & partners!

### LocalSolver in Japanese // 08 Oct 2012

LocalSolver now speaks Japanese thanks to our partner MSI.

### Logistics & Automation Consulting chooses LocalSolver // 24 Sep 2012

The Italian company Logistics & Automation Consulting (LAC) chooses LocalSolver as mathematical programming partner.

### LocalSolver now in OR education // 10 Sep 2012

LocalSolver appears as complement of classical MIP and CP solvers in OR courses in French universities and high schools.

### LocalSolver at ISMP 2012 // 15 Aug 2012

LocalSolver was at ISMP 2012, the world congress in mathematical programming, held in Berlin. Here are the (provocative!) slides.

### Customers & Partners // 02 Aug 2012

Welcome to our new customers & partners!

### New release: LocalSolver 2.1 // 06 Jul 2012

LocalSolver 2.1 was released the 6th July 2012. All users of LocalSolver 2.0 can take advantage of this new version for free.

### LocalSolver at EURO 2012 // 27 Jun 2012

LocalSolver is a sponsor of EURO 2012 in Vilnius (8-11 July). Come on exhibition booth #20 to discover LocalSolver and the team!

We have submitted a LocalSolver-based solution for the final round of Google ROADEF Challenge. Here are given our results.

### LocalSolver in Optimization Days 2012 // 04 May 2012

LocalSolver was presented in Optimization Days 2012 in Montreal. You can download the slides of the presentation.

### Frédéric Gardi wins the 2012 Robert Faure Prize // 16 Apr 2012

This prize is awarded every 3 years by the French OR society to a worthy young researcher. This is the first time it has been awarded to an engineer working in industry and not to an academic.

### LocalSolver at ROADEF 2012 // 08 Apr 2012

LocalSolver sponsorizes ROADEF 2012 congress. LocalSolver 2.0 will be presented in industrial semi-plenary session. Come to discover it!

### LocalSolver available on Mac // 26 Mar 2012

Good news for Apple fans: LocalSolver is now available on Mac.

### Official launch of LocalSolver 2.0 at Bouygues Headquarters // 11 Mar 2012

LocalSolver team invites you to the official launch of the commercial version of its new-generation solver: Tuesday March 13 2012.

### LocalSolver tackles B instances on Google ROADEF Challenge // 24 Feb 2012

LocalSolver is able to solve some of the ultra-large instances from set B. Here are given some results.