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Here, you can find several aspects of the solution of the model:

• The model overview page gives an overview of the model: what type of problem is it, how many variables does it have, and how many constraints? If the model is two-dimensional, a graph of the feasible region is displayed.
• The variables page lists the variables of the model, their corresponding optimal values, and several other values.
• The constraints page lists the constraints of the model, their corresponding optimal right hand side value, and several other values.
• Output contains any output generated by the model.
• Log messages contains (debugging) information provided by the solution algorithm.

## Graphical Method Calculator – Linear Programming 🥇

Do you often find yourself confused by linear programming problems that you can't solve? Maybe it's time to get some help. Well… You're in luck! Solving optimization exercises with the graphical method will be easier with our graphical method calculator for linear programming problems.

## Graphical Method Calculator - Linear Programming

Objective function:, constraints.

Constraint 1:

Constraint 2:

X 1 , X 2 ≥ 0

## Members-Only Content

Do you already have a membership, get membership.

The above application is a simplified version of our graphing method calculator available to students who have a membership with us; however, it has all the basic functionality required to graph most linear programming exercises in your school.

This free version, like other available free linear programming calculators, only shows the final result (optimal solution and graph) of the problem. Since many students cannot adequately understand how the graphs were generated, we have developed a version with detailed step-by-step explanations of the solution of the problem.

## Advanced Functions of the Graphical Method of Linear Programming Calculator

Our membership aims to help you improve your problem solving skills and perform better in your school. That is why we include a series of online resources, where linear programming is a must. In this application you will find the following:

• Calculation of the intersections with the axes to graph each constraint.
• Explanation of the area to shade depending on the type of inequality.
• Determination of the feasible region.
• Location of the objective function on the graph, if applicable.
• Determination of special cases such as unbounded, unbounded or infeasible solutions.
• Solve exercises with inequalities or equations.
• You can enter a maximum of 10 restrictions and 2 variables.

You can find complete examples of how the application works in this link .

## How to use the Online Graphical Method Calculator

The use of our calculator is very simple and intuitive, however, we will explain its use step by step:

• Before starting, you must have made the approach of the model to be optimized. Remember that for the graphical method we normally work with 2 decision variables.
• You must enter the coefficients of the objective function and the constraints. You can enter integer values, fractions and decimals. Likewise, you must also select the sign of the inequalities.
• To enter the coefficients of the objective function and the constraints, you can use integer values as well as fractions and decimals. You must also select the sign of the inequalities.
• Click on “Solve / Graph” .
• If you are in the free version, you will immediately get the final graph and results. In the full version, you will be able to see the step by step from the creation of the graphs to the final result.

Next we will see some images of the operation of the calculator:

This calculator facilitates your learning of the graphical method and combines well with our simplex method application (two phases) and our Big M Method calculator .

## Final Reflection

We know that the best way to learn something is to have the right tools to do it. At PM Calculators we are working to bring you the best tools gathered in one place. If you have any recommendation to improve our calculator, write us to our contact form.

• MAX Z = 100x1 + 150x2 subject to 8000x1 + 4000x2 15x1 + 30x2 and x1,x2 >= 0
• MAX Z = 7x1 + 9x2 subject to -x1 + 3x2 7x1 + x2 x2 and x1,x2 >= 0
• MAX Z = 3x1 + 5x2 subject to 2x1 + 4x2 x1 2x2 and x1,x2 >= 0
• =,>=20,50');">MIN Z = 3x1 + 2.5x2 subject to x1 + 2x2 >= 20 3x1 + 2x2 >= 50 and x1,x2 >= 0
• MAX Z = 2x1 + 3x2 subject to 4x1 + 12x2 10x1 + 4x2 and x1,x2 >= 0

Solving Constraint Integer Programs

## How to Cite

• Documentation

## SCIP Workshops

Related work, team members, cooperation, imprint and privacy statement.

SCIP is currently one of the fastest non-commercial solvers for mixed integer programming (MIP) and mixed integer nonlinear programming (MINLP). It is also a framework for constraint integer programming and branch-cut-and-price. It allows for total control of the solution process and the access of detailed information down to the guts of the solver.

older news...

## What is SCIP?

A similar technique is used for solving both Integer Programs and Constraint Programs: the problem is successively divided into smaller subproblems (branching) that are solved recursively.

On the other hand, Integer Programming and Constraint Programming have different strengths: Integer Programming uses LP relaxations and cutting planes to provide strong dual bounds, while Constraint Programming can handle arbitrary (non-linear) constraints and uses propagation to tighten domains of variables.

SCIP is a framework for Constraint Integer Programming oriented towards the needs of mathematical programming experts who want to have total control of the solution process and access detailed information down to the guts of the solver. SCIP can also be used as a pure MIP and MINLP solver or as a framework for branch-cut-and-price.

SCIP is implemented as C callable library and provides C++ wrapper classes for user plugins. It can also be used as a standalone program to solve mixed integer linear and nonlinear programs given in various formats such as MPS, LP, flatzinc, CNF, OPB, WBO, PIP, etc. Furthermore, SCIP can directly read ZIMPL models.

## Interfaces and LP solvers usable with SCIP

Detailed list of scip's features.

• very fast standalone solver for linear programming (LP), mixed integer programming (MIP), and mixed integer nonlinear programming (MINLP)
• framework for branching, cutting plane separation, propagation, pricing, and Benders' decomposition,
• constraint handlers to implement arbitrary constraints,
• variable pricers to dynamically create problem variables,
• domain propagators to apply constraint independent propagations on the variables' domains,
• separators for cutting planes based on the LP relaxation; benefit from a dynamic cut pool management,
• relaxators can be included to provide relaxations (e.g., semidefinite relaxations or Lagrangian relaxations) and dual bounds in addition to the LP relaxation, working in parallel or interleaved
• plugins to apply Benders' decomposition and implement Benders' cuts,
• primal heuristics to search for feasible solutions with specific support for probing and diving,
• node selectors to guide the search,
• branching rules to split the problem into subproblems; arbitrarily many children per node can be created, and the different children can be arbitrarily defined,
• presolvers to simplify the solved problem,
• file readers to parse different input file formats,
• event handlers to be informed on specific events, e.g., after a node was solved, a specific variable changes its bounds, or a new primal solution is found,
• display handlers to create additional columns in the solver's output.
• dialog handlers to extend the included command shell.
• conflict analysis can be applied to learn from infeasible subproblems
• dynamic memory management to reduce the number of operation system calls with automatic memory leakage detection in debug mode

## What is the SCIP Optimization Suite?

The SCIP Optimization Suite is a toolbox for generating and solving mixed integer nonlinear programs, in particular mixed integer linear programs, and constraint integer programs. It consists of the following parts:

The user can easily generate linear, mixed integer and mixed integer quadratically constrained programs with the modeling language ZIMPL. The resulting model can directly be loaded into SCIP and solved. In the solution process SCIP may use SoPlex as underlying LP solver.

Since all six components are available in source code and free for academic use, they are an ideal tool for academic research purposes and for teaching mixed integer programming.

A further extension of SCIP in order to solve MISDPs (mixed-integer semidefinite programs) is available from TU Darmstadt: SCIP-SDP .

SCIP also includes an extension for solving Steiner tree and related problems: SCIP-Jack .

Enabling Research through the SCIP Optimization Suite 8.0 Ksenia Bestuzheva, Mathieu Besançon, Wei-Kun Chen, Antonia Chmiela, Tim Donkiewicz, Jasper van Doornmalen, Leon Eifler, Oliver Gaul, Gerald Gamrath, Ambros Gleixner, Leona Gottwald, Christoph Graczyk, Katrin Halbig, Alexander Hoen, Christopher Hojny, Rolf van der Hulst, Thorsten Koch, Marco Lübbecke, Stephen J. Maher, Frederic Matter, Erik Mühmer, Benjamin Müller, Marc E. Pfetsch, Daniel Rehfeldt, Steffan Schlein, Franziska Schlösser, Felipe Serrano, Yuji Shinano, Boro Sofranac, Mark Turner, Stefan Vigerske, Fabian Wegscheider, Philipp Wellner, Dieter Weninger, Jakob Witzig Available at ACM Digital Library , June 2023 BibTeX

The SCIP Optimization Suite 8.0 Ksenia Bestuzheva, Mathieu Besançon, Wei-Kun Chen, Antonia Chmiela, Tim Donkiewicz, Jasper van Doornmalen, Leon Eifler, Oliver Gaul, Gerald Gamrath, Ambros Gleixner, Leona Gottwald, Christoph Graczyk, Katrin Halbig, Alexander Hoen, Christopher Hojny, Rolf van der Hulst, Thorsten Koch, Marco Lübbecke, Stephen J. Maher, Frederic Matter, Erik Mühmer, Benjamin Müller, Marc E. Pfetsch, Daniel Rehfeldt, Steffan Schlein, Franziska Schlösser, Felipe Serrano, Yuji Shinano, Boro Sofranac, Mark Turner, Stefan Vigerske, Fabian Wegscheider, Philipp Wellner, Dieter Weninger, Jakob Witzig Available at Optimization Online and as ZIB-Report 21-41 , December 2021 BibTeX

The SCIP Optimization Suite 7.0 Gerald Gamrath, Daniel Anderson, Ksenia Bestuzheva, Wei-Kun Chen, Leon Eifler, Maxime Gasse, Patrick Gemander, Ambros Gleixner, Leona Gottwald, Katrin Halbig, Gregor Hendel, Christopher Hojny, Thorsten Koch, Pierre Le Bodic, Stephen J. Maher, Frederic Matter, Matthias Miltenberger, Erik Mühmer, Benjamin Müller, Marc Pfetsch, Franziska Schlösser, Felipe Serrano, Yuji Shinano, Christine Tawfik, Stefan Vigerske, Fabian Wegscheider, Dieter Weninger, Jakob Witzig Available at Optimization Online and as ZIB-Report 20-10 , March 2020 BibTeX

The SCIP Optimization Suite 6.0 Ambros Gleixner, Michael Bastubbe, Leon Eifler, Tristan Gally, Gerald Gamrath, Robert Lion Gottwald, Gregor Hendel, Christopher Hojny, Thorsten Koch, Marco E. Lübbecke, Stephen J. Maher, Matthias Miltenberger, Benjamin Müller, Marc E. Pfetsch, Christian Puchert, Daniel Rehfeldt, Franziska Schlösser, Christoph Schubert, Felipe Serrano, Yuji Shinano, Jan Merlin Viernickel, Matthias Walter, Fabian Wegscheider, Jonas T. Witt, Jakob Witzig Available at Optimization Online and as ZIB-Report 18-26 , July 2018 BibTeX

The SCIP Optimization Suite 5.0 Ambros Gleixner, Leon Eifler, Tristan Gally, Gerald Gamrath, Patrick Gemander, Robert Lion Gottwald, Gregor Hendel, Christopher Hojny, Thorsten Koch, Matthias Miltenberger, Benjamin Müller, Marc E. Pfetsch, Christian Puchert, Daniel Rehfeldt, Franziska Schlösser, Felipe Serrano, Yuji Shinano, Jan Merlin Viernickel, Stefan Vigerske, Dieter Weninger, Jonas T. Witt, Jakob Witzig Available at Optimization Online and as ZIB-Report 17-61 , December 2017 BibTeX

The SCIP Optimization Suite 4.0 Stephen J. Maher, Tobias Fischer, Tristan Gally, Gerald Gamrath, Ambros Gleixner, Robert Lion Gottwald, Gregor Hendel, Thorsten Koch, Marco E. Lübbecke, Matthias Miltenberger, Benjamin Müller, Marc E. Pfetsch, Christian Puchert, Daniel Rehfeldt, Sebastian Schenker, Robert Schwarz, Felipe Serrano, Yuji Shinano, Dieter Weninger, Jonas T. Witt, Jakob Witzig Available at Optimization Online and as ZIB-Report 17-12 , March 2017 BibTeX

The SCIP Optimization Suite 3.2 Gerald Gamrath, Tobias Fischer, Tristan Gally, Ambros M. Gleixner, Gregor Hendel, Thorsten Koch, Stephen J. Maher, Matthias Miltenberger, Benjamin Müller, Marc E. Pfetsch, Christian Puchert, Daniel Rehfeldt, Sebastian Schenker, Robert Schwarz, Felipe Serrano, Yuji Shinano, Stefan Vigerske, Dieter Weninger, Michael Winkler, Jonas T. Witt, Jakob Witzig Available at Optimization Online and as ZIB-Report 15-60 , February 2016 BibTeX

In order to reference the general algorithmic design behind constraint integer programming and SCIP's solving techniques regarding mixed-integer linear and nonlinear programming, please cite the following articles:

• Constraint Integer Programming: a New Approach to Integrate CP and MIP Tobias Achterberg, Timo Berthold, Thorsten Koch, Kati Wolter Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, CPAIOR 2008, LNCS 5015, Pages 6–20, 2008 Available as ZIB Report 08-01 and at SpringerLink , January 2008 BibTeX
• SCIP: Solving Constraint Integer Programs Tobias Achterberg Mathematical Programming Computation, Volume 1, Number 1, Pages 1–41, 2009. Available at Mathematical Programming Computation , 2009 BibTeX

A more detailed description of SCIP can be found in

• Constraint Integer Programming Tobias Achterberg Ph.D. thesis, TU Berlin, July 2007. Available as Doctoral Thesis , January 2009 BibTeX

The features for the global optimization of convex and nonconvex MINLPs are described in

• Global Optimization of Mixed-Integer Nonlinear Programs with SCIP 8 Ksenia Bestuzheva, Antonia Chmiela, Benjamin Müller, Felipe Serrano, Stefan Vigerske, Fabian Wegscheider arXiv:2301.00587, 2023. BibTeX

The extension of SCIP to solve MIPs exactly over rational input data is described in

• A Computational Status Update for Exact Rational Mixed Integer Programming Leon Eifler, Ambros Gleixner Integer Programming and Combinatorial Optimization. IPCO 2021, Lecture Notes in Computer Science, vol 12707. pp. 163-177 BibTeX

However, for the latest developments, please consult our series of release reports .

Releases up to and including Version 8.0.2 remain under the ZIB Academic License as indicated by the files contained in the releases. The new license applies from Version 8.0.3 onwards.

Please note that version x.x.x is currently in beta stage.

## Source Code

You can either download SCIP alone or the SCIP Optimization Suite (recommended), a complete source code bundle of SCIP, SoPlex, ZIMPL, GCG, PaPILO and UG.

• For compilation instructions please consult the installations section of the online documentation.
• SCIP is completely implemented in C. The code should compile with any C compiler that supports the C99 standard. We have tested SCIP with GNU, Clang and Intel compilers on 32- and 64-bit versions of Linux, Mac and Windows.
• If you are using the GNU compiler on SunOS and you experience a strange behavior of your program (segmentation faults), you might try a reduce the optimization level -O2 . If problems occur with STL code, you might change to a different implementation by adding -library=stlport4 to CXX_COMPILER_FLAGS . (Note: There are different implementations of the STL on SUN platforms.)

## Precompiled Packages

You can also download precompiled executables of SCIP with which you can solve MIP, MIQCP, CIP, SAT, or PBO instances in MPS, LP, RLP, ZIMPL, flatzinc, CNF, OPB, WBO, PIP, or CIP format. Note that these executables do not include the readline features (i.e., command line editing and history) due to license issues. However, you can download the free readline wrapper rlwrap to provide this missing feature to the executables.

If you use conda , you can install the components of the scipoptsuite using the conda packages .

• In order to build SCIP yourself with support for symmetry handling , you need bliss . A patched bliss fork is now available on GitHub . This fork allows to limit the number of search nodes and generators during the backtrack search and improves performance of symmetry handling in SCIP.
• For some of the mac packages you need the libgfortran and libtbb libraries, and bison . You can install them for example via homebrew: brew install bison, brew install gcc and brew install tbb before installing the scipoptsuite. After installation of the .sh package to a custom directory you may need to include the lib folder into your DYLD_LIBRARY_PATH. If you want to compile the Suite on a apple silicon M series processor you might need to set ARCH=arm as an argument to make.
• The linux precompiled binaries are built on debian and ubuntu, both debian based distributions. Chances are that you won't be able to install them on a different one, like arch-linux.
• If you download an executable containing PaPILO you might need a working installation of the TBB library in your machine. (This might also be available via your package manager, for debian, for example, it is called libtbb2.)
• For GCG on a linux system you need a working installation of hMETIS version >= 2.0, and zsh (apt install zsh) in your system.
• Here is the download section of MIPLIB 2017 benchmark MPS files.
• SCIP is also available on the NEOS Server , where you can post your model in LP or MPS format, or as an AMPL, GAMS, or ZIMPL model and let the NEOS Server solve it with SCIP linked to CPLEX.
• A development version of SCIP 7 to solve MIPs exactly without floating-point roundoff errors is available as a separate download on Github .

Every now and then there are SCIP Workshops where developers and users come together to discuss different approaches and implementations.

The next workshop will be held on the occasion of SCIP's 20th anniversary on November 4th, 2022. Click here for further information. Sign up for the SCIP mailing list to get notified about future events.

• An up-to-date list of publications about SCIP can be found at swMath.org .

## Projects at ZIB that use SCIP

• Optimization of Gas Transport ,DFG Research Center Matheon , Project B20.
• Advanced Solver Technology for SCM
• DESI – Durchgängig energiesensible IKT-Produktion
• Exact Integer Programming , DFG Priority Program 1307 Algorithm Engineering .
• Integrated Planning of Multi-layer Networks , DFG Research Center Matheon ,  Project B3.
• Service Design in Public Transport , DFG Research Center Matheon ,  Project B15.
• ForNe: Research Cooperation Network Optimization
• Chip Design Verification , DFG Research Center Matheon ,  Project D17.
• Discrete Morse Functions .
• Infeasible Linear Inequality Systems .
• MLTN - Multi-layer Transport Networks
• Stable sets and special graph classes
• Symmetries in Integer Programming , DFG Research Center Matheon ,  Project B12.
• VeriCount - Counting Solutions in the Field of Verification
• Scheduling the SBB Cargo Railroad routing and shipment operations at night, Combinatorial Optimization & Graph Algorithms Group , TU Berlin.
• Scheduling Techniques in Constraint Integer Programming, DFG Research Center Matheon ,  Project B25.

## Projects using SCIP (outside ZIB)

• Project A9: Resilient Design from SFB 805: Control of Uncertainties in Load-Carrying Structures in Mechanical Engineering , TU Darmstadt
• Projekt EXPRESS: Exploiting Structure in Compressed Sensing Using Side Constraints from SPP 1798: Compressed Sensing in Information Processing (CoSIP) , TU Darmstadt
• Project A01: Global Methods for Stationary Gastransport from Transregio/SFB 154: Mathematical Modelling, Simulation and Optimization on the Example of Gas Networks , TU Darmstadt
• Project A4: Mathematical Models and Methods for an Optimum Combination of Passive and Active Components from SFB 805: Control of Uncertainties in Load-Carrying Structures in Mechanical Engineering , TU Darmstadt
• DSP – Parallel Solver for Stochastic Mixed-integer Programming Problems, Argonne National Laboratory
• GOBNILP – Globally Optimal Bayesian Network learning using Integer Linear Programming, The University of York
• SCIL – Symbolic Constraints in Integer Linear programming , Max-Planck-Institut für Informatik
• Generic Column Generation , RWTH Aachen
• Normaliz , a tool for computations in affine monoids, vector configurations, lattice polytopes, and rational cones developed at the University of Osnabrück.
• Numberjack – A python constraint programming platform, University College Cork
• Thermos Map-driven web-based software for optimising the layout of heat networks, and associated applications, Centre for Sustainable Energy, UK

## Some Papers that use SCIP

• Conflict Analysis in Mixed Integer Programming Tobias Achterberg Discrete Optimization, Special Issue 4, 2007
• Hybrid Branching Tobias Achterberg, Timo Berthold Proc. of CPAIOR 2009, LNCS 5547, 05.2009.
• Improving the Feasibility Pump Tobias Achterberg, Timo Berthold Discrete Optimization, Special Issue 4, 2007
• Constraint Integer Programming: Techniques and Applications Tobias Achterberg, Timo Berthold, Stefan Heinz, Thorsten Koch, Kati Wolter ZIB-Report 08-43.
• Constraint Integer Programming: a New Approach to Integrate CP and MIP Tobias Achterberg, Timo Berthold, Thorsten Koch, Kati Wolter Proc. of CPAIOR 2008, LNCS 5015, 2008
• Teaching MIP Modeling and Solving Tobias Achterberg, Thorsten Koch, and Martin Grötschel ORMS Today 33, no. 6
• Counting solutions of integer programs using unrestricted subtree detection Tobias Achterberg, Stefan Heinz, Thorsten Koch Proc. of CPAIOR 2008, LNCS 5015, 2008
• Experiments with Linear and Semidefinite Relaxations for Solving the Minimum Graph Bisection Problem Michael Armbruster, Marzena Fügenschuh, Christoph Helmberg, and Alexander Martin Technical Report, TU Darmstadt, 2006
• Constrained Clustering using Column Generation Babaki, B., Guns, T., Nijssen, S. CPAIOR, May 2014, Cork, Ireland
• Heuristics of the Branch-Cut-and-Price-Framework SCIP Timo Berthold Operations Research Proceedings 2007
• RENS - Relaxation Enforced Neighborhood Search Timo Berthold ZIB-Report 07-28
• Rapid learning for binary programs Timo Berthold, Thibaut Feydy, Peter J. Stuckey Proc. of CPAIOR 2010, LNCS 6140
• SCIP Optimization Suite を利用した 混合整数(線形/非線形) 計画問題の解法 Timo Berthold, Ambros Gleixner, Stefan Heinz, Thorsten Koch, and Yuji Shinano. ZIB-Report 12-24
• Undercover – a primal heuristic for MINLP based on sub-MIPs generated by set covering Timo Berthold, Ambros M. Gleixner ZIB-Report 09-40
• A Constraint Integer Programming Approach for Resource-Constrained Project Scheduling Timo Berthold, Stefan Heinz, Marco Lübbecke, Rolf H. Möhring, Jens Schulz Proc. of CPAIOR 2010, LNCS 6140
• Nonlinear pseudo-Boolean optimization: relaxation or propagation? Timo Berthold, Stefan Heinz, Marc E. Pfetsch Theory and Applications of Satisfiability Testing – SAT 2009, LNCS 5584, 2009
• Solving Pseudo-Boolean Problems with SCIP Timo Berthold, Stefan Heinz, Marc E. Pfetsch ZIB-Report 08-12
• Extending a CIP framework to solve MIQCPs Timo Berthold, Stefan Heinz, Stefan Vigerske ZIB-Report 09-23
• Comparing MIQCP solvers to a specialised algorithm for mine production scheduling Andreas Bley, Ambros M. Gleixner, Thorsten Koch, Stefan Vigerske ZIB-Report 09-32
• Auslegung heterogener Kommunikationsnetze nach Performance und Wirtschaftlichkeit Andreas Bley, Friederich Kupzog, and Adrian Zymolka Proc. of the 11th Kasseler Symposium Energie-Systemtechnik: Energie und Kommunikation, 2006
• Angebotsplanung im öffentlichen Nahverkehr Ralf Borndörfer, Marika Neumann, Marc E. Pfetsch ZIB-Report 08-04, to appear in Heureka'08
• The Location-Dispatching Problem: polyhedral results and Content Delivery Network Design Philippe Chrétienne, Pierre Fouilhoux, Eric Gourdin, and Jean-Mathieu Segura Electronic Notes in Discrete Mathematics 26, 867-874, 2010
• Coordination of Cluster Ensembles via Exact Methods Ioannis T. Christou IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(2):279—293, 2011
• A Branch-and-Price Algorithm for Multi-mode Resource Leveling Eamonn T. Coughlan, Marco E. Lübbecke and Jens Schulz Proc. of SEA 2010, LNCS 6049, 226-238, 2010
• Models and Algorithms for Maximum Flow Problems Having Semicontinuous Path Flow Constraints Robert Curry and J. Cole Smith IISE Transactions, 2017
• Optimal control of spatial-dynamic processes: the case of biological invasions Rebecca S. Epanchin-Niell and James E. Wilen Agricultural and Applied Economics Association 2010 Annual Meeting
• Integer linear programming models for topology optimization in sheet metal design Armin Fügenschuh and Marzena Fügenschuh Mathematical Methods of Operations Research, to appear (2008)
• Scenario Technique with Integer Programming for Sustainability in Manufacturing Armin Fügenschuh, Pia Gausemeier, Günther Seliger, and Semih Severengiz Lecture Notes in Business Information Processing 46, 320-331, 2010
• Experiments with a Generic Dantzig-Wolfe Decomposition for Integer Programs Gerald Gamrath and Marco E. Lübbecke Proc. of SEA 2010, LNCS 6049, 239-252, 2010
• Ein neuer Ansatz zur Optimierung des Bilanzausgleichs in einem Gasmarktgebiet Uwe Gotzes Zeitschrift für Energiewirtschaft, 1-13, 2019
• Using Model Counting to Find Optimal Distinguishing Tests Stefan Heinz and Martin Sachenbacher Proc. of CPAIOR 2009, LNCS 5547
• Exact and Approximate Sparse Solutions of Underdetermined Linear Equations Sadegh Jokar, Marc E. Pfetsch ZIB-Report 07-05
• Computing Optimal Morse Matchings Michael Joswig and Marc E. Pfetsch SIAM Journal on Discrete Mathematics 20, no. 1, 2006
• Orbitopal Fixing Volker Kaibel, Matthias Peinhardt, and Marc E. Pfetsch Proc. of the 12th Integer Programming and Combinatorial Optimization conference (IPCO) M. Fischetti and D. Williamson (eds.), LNCS 4513, Springer-Verlag, 74-88, 2007
• On connectivity limits in ad hoc networks with beamforming antennas Moritz Kiese, Christian Hartmann, Julian Lamberty, and Robert Vilzmann EURASIP Journal on Wireless Communications and Networking 2009, No.7, 74-88, 02.2009
• Approximated segmentation considering technical and dosimetric constraints in intensity-modulated radiation therapy with electrons Antje Kiesel and Tobias Gauer ArXiv e-prints, May 2010
• Rapid Mathematical Programming or How to Solve Sudoku Puzzles in a few Seconds Thorsten Koch Operations Research Proceedings 2005
• Algorithms to separate {0,1/2}-Chvatal-Gomory cuts Arie M. C. A. Koster, Adrian Zymolka and Manuel Kutschka Algorithmica 55, No. 2, 375-391
• A formulation space search heuristic for packing unequal circles in a fixed size circular container C. O. López and J. E. Beasley European Journal of Operational Research 251 (2016) 64–73
• Large-scale identification of genetic design strategies using local search Desmond S. Lun, Graham Rockwell, Nicholas J. Guido, Michael Baym, Jonathan A. Kelner, Bonnie Berger, James E. Galagan, and George M. Church Molecular Systems Biology 5, No. 296, 2009
• Optimization Methods For Selecting Founder Individuals For Captive Breeding or reintroduction of Endangered Species Webb Miller, Stephen J. Wright, Yu Zhang, Stephan C. Schuster, Vanessa M. Hayes Pacific Symposium on Biocomputing, 43-53, 2010
• Two-layer Network Design by Branch-and-Cut featuring MIP-based Heuristics Sebastian Orlowski, Arie M. C. A. Koster, Christian Raack, and Roland Wessäly Proceedings of the Third International Network Optimization Conference, 2007
• Branch-And-Cut for the Maximum Feasible Subsystem Problem Marc E. Pfetsch SIAM Journal on Optimization 19, No. 1, 21-38 (2008)
• Integer Programming and Sports Rankings Christian Raack, Annie Raymond, Thomas Schlechte, Axel Werner ZIB-Report 13-19
• Rostering from staffing levels: a branch-and-price approach Egbert van der Veen and Bart Veltman Proc. of the 35th International Conference on Operational Research Applied to Health Services, 1-10, 2009
• Faster MIP solutions via new node selection rules Daniel T. Wojtaszeka and John W. Chinneck Computers & Operations Research 37, 1544-1556, September 2009
• Optimal Packings of Congruent Circles on a Square Flat Torus as Mixed-Integer Nonlinear Optimization Problem. Vladimir Voloshinov and Sergey Smirnov Voevodin V., Sobolev S. (eds) Supercomputing. RuSCDays 2019. Communications in Computer and Information Science, vol 1129. Springer, December 2019
• Robust Aircraft Routing Chiwei Yan and Jerry Kung INFORMS Transportation Science, July 2016

If you know about further projects or papers that use SCIP, please let us know .

For general information or questions about SCIP please write to the SCIP mailing list [email protected] after subscribing to it at the SCIP mailing list page . For licensing questions, please see the license section of the web page and the contact provided there. Trouble compiling SCIP from source? Please check the build documentation before sending an email. For questions about our SCIP interfaces on GitHub please open an issue in the corresponding repository.

## Mailing List

The SCIP mailing list can be accessed via the SCIP mailing list page . You can conveniently search the archives using Google: site:listserv.zib.de/pipermail/scip

## Stack Overflow

We are also watching the SCIP tag on stackoverflow.com and will answer your questions there. Note that we will not answer faster only because you posted the same question both to stack overflow and the mailing list.

## Reporting Bugs

SCIP has more than 500,000 lines of source code and is definitely not bug free. If you'd like to help us improve SCIP, visit our bug submission page and file a bug report in English or German.

## Student Assistants

We are thankful to many people who over the years have contributed code to SCIP, among others:

SCIP is developed in cooperation with

© 2024 by Zuse Institute Berlin (ZIB). For the imprint and privacy statement we refer to the Imprint of ZIB with the following additions and modifications:

## Generating PDF...

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• The utility is quite flexible with input. For instance, the following format will also be accepted (inequalities separated by commas): Maximize p = x+y subject to x+y = 4
• However, the numbers on the right-hand sides of all constraints must be non-negative.
• Do not use commas in large numbers. For instance, enter 100,000 as 100000.
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## What Is the Linear Programming Calculator?

What is linear programming (lp), representing linear programs.

This online calculator solves the problem of finding the correct solution or optimized output of the desired mathematical models by providing a quick, reliable, and accurate solution.

It just requires the user to enter the objective function along with the system of linear constraints and the solution will be on their screens just in a matter of seconds. The Linear Programming Calculator is the most efficient tool for linear optimization and can be used to solve complex and time-consuming problems and models effectively and logically.

The Linear Programming Calculator is an online calculator that can be used for the linear optimization of various mathematical models.

It is a convenient and user-friendly tool with an easy-to-use interface that helps the user to find the exact and optimized solution for the provided constraints faster than any other mathematical technique applied manually.

The Linear Programming Calculator helps the user to avoid the long mathematical calculations and get the desired answer just by clicking one button.

The calculator can solve problems containing a maximum of nine different variables not more than that. It requires “ , ” as a separator for multiple constraints in a single box.

Let’s find out more about the calculator and how it works.

## How To Use a Linear Programming Calculator?

You can use the Linear Programming Calculator by entering the objective function and specifying the constraints. Once you are done entering all the inputs you just have to press the submit button and a detailed solution will be displayed on the screen just in seconds.

Following are the detailed stepwise guidelines to find out the best possible solution for the given objective function with specified constraints. Follow these simple steps and find out the maxima and minima of the functions.

Consider your desired objective function and specify its constraints.

Now, enter the objective function in the tab specified as Objective Function .

After adding the objective function, input the conditions of all constraints in the tab named Subject . The calculator can take a maximum of nine constraints and has more tabs for it under the name More Constraints . To add multiple constraints in a single block, you have to use “,” as a separator.

Once you are done filling all the input fields, select the optimization category from the Optimize drop-down menu. There are three options you can select to find the maxima of the objective function, minima of the objective function or you can select both.

The options in the drop-down menu are given as:

After that, press the Submit button and the optimal solution along with graphs will be displayed in the result window.

Make sure not to add more than nine constraints in the calculator, otherwise it will fail to produce the desired results.

You can view the result window below the calculator layout. The Result window contains the following blocks:

## Input Interpretation

This block shows the input entered by the user and how it has been interpreted by the calculator. This block helps the user to figure out if there were any mistakes in the input data.

## Global Maximum

This block shows the calculated global maxima of the given objective function. Global maxima are the overall largest value of the objective function.

## Global Minimum

This block displays the global minima of the given objective function. Global minima are the overall smallest value of the given function with the specified constraints.

This block displays the 3D interpretation of the objective function. It also specifies the maxima and minima points on the 3D plot.

## Contour Plot

The contour plot is a 2D representation of the global maxima and global minima of the objective function on the graph.

## How Does the Linear Programming Calculator Work?

The Linear Programming Calculator works by computing the best optimal solution of the objective function using the technique of Linear programming, which is also called Linear optimization.

Mathematical optimization is the technique used to find the best possible solution to a mathematical model such as finding the maximum profit or analyzing the size of the cost of a project, etc. It is the type of linear programming that helps to optimize the linear function provided that given constraints are valid.

To understand more about the working of the Linear Programming Calculator , let’s discuss some of the important concepts involved.

Linear Programming is the mathematical programming technique that tends to follow the best optimal solution of a mathematical model under specified conditions which are called constraints. It takes various inequalities applied to a certain mathematical model and finds the optimum solution.

Linear Programming is only subjected to linear equality and inequality constraints. It is only applicable to linear functions that are the first-order functions. The linear function is usually represented by a straight line and the standard form is y = ax + b.

In linear programming , there are three components: decision variables, objective function, and constraints. The usual form of a linear program is given as follows:

The first step is to specify the decision variable that is an unknown element in the problem.

decision variable = x

Then, decide whether the optimization required is the maximum value or the minimum value.

The next step is to write the objective function that can be maximized or minimized. The objective function can be defined as:

$X \to C^T \times X$

Where  C is the vector.

Finally, you need to describe the constraints that can be in the form of equalities or inequalities and they must be specified for the given decision variables.

The constraints for the objective function can be defined as:

$AX \leq B$

$X \geq 0$

Where A and B are the vectors. Therefore, linear programming is an effective technique for the optimization of various mathematical models.

Thus, the Linear Programming Calculator uses the linear programming process to solve the problems in seconds.

Due to its effectiveness, it can be utilized in various fields of study. Mathematicians and businessmen use it widely, and it is a very useful tool for engineers to help them solve complex mathematical models that are formed for various designing, planning, and programming purposes.

A linear program can be represented in various forms. First, it requires the identification of the maximization or minimization of the objective function and then the constraints. The constraints can be either in the form of inequalities $( \leq , \geq )$ or equality ( = ).

A linear program can have decision variables represented as $x_1, x_2, x_3, …….., x_n$.

Therefore, the general form of a Linear Program is given as:

Minimize or Maximize:

$y = c_o + c_1x_1 + c_2x_2 + …. + c_nx_n$

Subject to:

$a_1i x_1+ a_2ix_2 + a_3ix_3 +……. + a_nix_n = b_i$

$a_1ix_1 + a_2ix_2 + a_3ix_3 +……. + a_nix_n \leq b_i$

$a_1ix_1+ a_2ix_1 + a_3ix_2 +……. + a_nix_n \geq b_i$

Where  i = 1,2,3,……..,m.

$x_k \geq 0$

$x_k < 0$

$x_k > 0$

Where k = 1,2,3,……..,m.

Here $x_k$ is the decision variable and $a_in$, $b_i$, and $c_i$ are the coefficients of objective function.

## Solved Examples

Let’s discuss some examples of linear optimization of the mathematical problems using the Linear Programming Calculator.

Maximize and minimize the objective function given as:

$50x_1 + 40x_2$

The constraints for the above-mentioned objective function are given as:

$3x_1 + 1x_2 <= 2700$

$6x_1 + 4x_2 >= 600$

$5x_1 + 5x_2 = 600$

$x_1 \geq 0$

$x_2 \geq 0$

Use the calculator to optimize the given function.

Select the max/min option from the Optimize drop-down menu.

Input the objective function and the functional constraints in the specified blocks.

Now click the submit button to view the results.

The Global maximum of the function is given as:

$max( 50x_1 + 40x_2 )_{at ( x_1, x_2 )} = (120, 0 )$

The Global minimum of the function is given as:

$min ( 50x_1 + 40x_2 )_{at ( x_1, x_2 )} = (60, 60 )$

The 3D plot is shown in Figure 1:

The contour plot is given in Figure 2 below:

A diet plan chalked by the dietitian contains three types of nutrients from two types of food categories. The nutritional contents under study include proteins, vitamins, and starch. Let the two food categories be $x_1$ and $x_2$.

A specific amount of each nutrient must be consumed each day. The nutritional content of proteins, vitamins, and starch in food $x_1$ is 2, 5, and 7, respectively. For food category $x_2$ the nutritional contents of proteins, vitamins, and starch are 3,6, and, 8, respectively.

The requirement per day of each nutrient is 8, 15, and 7, respectively.

The cost of each category is $2$ per $kg$. Determine the objective function and constraints to find out how much food must be consumed per day to minimize the cost.

The decision variable are $x_1$ and $x_2$.

The objective function is given as:

$y = 2x_1 + 2x_2$

The various constraints for the given objective function analyzed from the data given above are:

$2x_1 + 3x_2 > 8$

$5x_1 + 6x_2 > 15$

$7x_1 + 8x_2 > 7$

All the constraints are non-negative as the amount of food cannot be negative.

Input all the data in the calculator and press the submit button.

Following results are obtained:

## Local Minimum

\[ min( 2x_1 + 2x_2 ) = (0, 2.67)

The 3D representation is shown in figure 3 below:

The contour plot is shown in Figure 4:

All the Mathematical Images/Graphs are created using GeoGebra.

## Exploring Mixed-Integer and Linear Programming Solvers

We know there are a range of solvers, free and paid, to choose from. We also know that for some situations, a free solver might be all that you need. This article is designed to help you better understand your choices among free solvers, their relative performance, and some questions to ask yourself in deciding what type of solver is right for you.

## Common Free Mixed-Integer and Linear Programming Solvers

There are several open-source packages available for optimization. Two popular ones are GLPK and LP_Solve. The nice thing about these packages is that it’s easy to get started because you can download and install them immediately. But these packages are far behind the state of the art in optimization.

## Open-Source Performance: Mixed-Integer and Linear Programming Comparisons

Performance is typically a crucial consideration when choosing a solver. To give a sense of the relative performance of the various solver options listed above, we’ve summarized the results of independent benchmark tests maintained by Hans Mittelmann at Arizona State.

If we look at performance on Mixed Integer Programming (MIP) models across a broad set of test models, the table below shows results along two key dimensions: a) was the solver able to solve the model, and b) how quickly was the model solved? As you can see from the results, performance varies widely across solvers.

Internally, when testing GLPK and LP Solve on over 2400 models (ones that Gurobi can solve in under 1000 seconds), one of the solvers failed in 63% of the test cases, while the other failed in about 72% of the test cases.

What does it mean to fail? Sometimes the free solvers simply reached the 1,002nd time limit before optimizing the problem. But in some cases, the solvers reported an incorrect solution or falsely concluded that the problem was infeasible .

People often assume that open-source packages are close to the performance of commercial products, figuring that the free solvers might be two times or maybe ten times slower than their commercial alternatives. The reality is that Gurobi can often be 10,000 times faster than open-source solvers.

Unfortunately, when people experience slow performance with free solvers, they often give up on optimization technology altogether .

## Three Reasons to Choose a Free Solver

As you can see from the data above, free solvers tend to struggle with practical models, either failing to solve them at all or solving them relatively slowly. However, we don’t mean to give you the impression that free solvers are never the right choice. Below are a few scenarios where you may want to consider a free solver.

## 1. There is no approved budget

Often times, when a company is first looking at using an optimization solver in their business, there may not be an approved budget. Management may still be trying to determine the role optimization can play in planning and decision-making, and the team doing the work is still “getting their feet wet.”

## 2. You prefer to program in C or C++

Most free solvers are written in C or C++ and don’t offer other APIs. GLPK requires you to use C. Gurobi offers both C and C++ APIs, as well as a full range of other APIs, including Python.

## 3. The model(s) being solved are both small and relatively easy to solve

Free solvers tend to struggle with larger and more difficult models, but if a free solver is able to solve your problem now, and you are confident that your problem won’t become more difficult in the future, then a free solver could be a reasonable choice.

We want to stress that you should consider future growth in the model. We’ve seen many situations where free solvers worked well on a small prototype but were unable to handle the production model. This can cause a lot of rework that you may be able to avoid.

Before you make the choice to use a free solver, we suggest that you look at relative performance on models of similar size and complexity to those that you are likely to want to solve eventually. We are happy to assist if you like. We have a very large library of models and may be able to run a specific comparison.

## Avoiding the “False Negative” Trap

One of the most important questions people tend to ask when they are first exploring solvers is if optimization is a fit for their business. We have seen cases where someone selected a free solver, tried building a model, and the solver just couldn’t handle the problem. As a result, they assumed their problem was just too complex to use optimization techniques.

Others find free solvers just too hard to use, given the solver didn’t support the programming language they preferred, they couldn’t get support, etc. The team then ends the project and moves on. This is unfortunate since, with the right tools and support, the project might have been a great success. If you find yourself in this situation, please contact us. We may be able to help steer you in the right direction so you get the results you need to support continuing with the project.

## Should I Start with Open-Source Tools and Switch to a Commercial Tool Later ?

It’s great that it’s easy to get started with open-source tools, but you should approach them with your eyes open. For example, if an open-source tool is unable to solve your model, you shouldn’t assume that your project is hopeless.

Additionally, a state-of-the-art commercial solver can save valuable development time in several ways. First, the interfaces in commercial solvers are typically more polished and advanced than those in open-source tools. For example, the Gurobi Python interface brings a modeling syntax to the popular Python language. Second, commercial systems like Gurobi offer pre-built IT infrastructure like client-server options and cloud computing.

## Overview of Free vs. Commercial Solvers

There are several important differences between free and commercial solvers you should keep in mind:

Sign up below for a free commercial evaluation or academic license. Then take Gurobi for a spin! You can even export and run your MPS files  and experience the difference for yourself.

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Mixed-integer linear programming (MILP)

## Description

Mixed-integer linear programming solver.

Finds the minimum of a problem specified by

min x f T x  subject to  { x ( intcon )  are integers A ⋅ x ≤ b A e q ⋅ x = b e q l b ≤ x ≤ u b .

f , x , intcon, b , beq , lb , and ub are vectors, and A and Aeq are matrices.

You can specify f , intcon, lb , and ub as vectors or arrays. See Matrix Arguments .

intlinprog applies only to the solver-based approach. For a discussion of the two optimization approaches, see First Choose Problem-Based or Solver-Based Approach .

x = intlinprog( f , intcon , A , b ) solves min f'*x such that the components of x in intcon are integers, and A*x ≤ b .

x = intlinprog( f , intcon , A , b , Aeq , beq ) solves the problem above while additionally satisfying the equality constraints Aeq*x = beq . Set A = [] and b = [] if no inequalities exist.

x = intlinprog( f , intcon , A , b , Aeq , beq , lb , ub ) defines a set of lower and upper bounds on the design variables, x , so that the solution is always in the range lb ≤ x ≤ ub . Set Aeq = [] and beq = [] if no equalities exist.

x = intlinprog( f , intcon , A , b , Aeq , beq , lb , ub , x0 ) optimizes using an initial feasible point x0 . Set lb = [] and ub = [] if no bounds exist.

x = intlinprog( f , intcon , A , b , Aeq , beq , lb , ub , x0 , options ) minimizes using the optimization options specified in options . Use optimoptions to set these options. Set x0 = [] if no initial point exists.

x = intlinprog( problem ) uses a problem structure to encapsulate all solver inputs. You can import a problem structure from an MPS file using mpsread . You can also create a problem structure from an OptimizationProblem object by using prob2struct .

[ x , fval , exitflag , output ] = intlinprog( ___ ) , for any input arguments described above, returns fval = f'*x , a value exitflag describing the exit condition, and a structure output containing information about the optimization process.

collapse all

## Solve an MILP with Linear Inequalities

Solve the problem

min x 8 x 1 + x 2 s u b j e c t t o { x 2 i s a n i n t e g e r x 1 + 2 x 2 ≥ - 1 4 - 4 x 1 - x 2 ≤ - 3 3 2 x 1 + x 2 ≤ 2 0 .

Write the objective function vector and vector of integer variables.

Convert all inequalities into the form A*x <= b by multiplying “greater than” inequalities by -1 .

Call intlinprog .

## Solve an MILP with All Types of Constraints

min x ( - 3 x 1 - 2 x 2 - x 3 ) s u b j e c t t o { x 3 b i n a r y x 1 , x 2 ≥ 0 x 1 + x 2 + x 3 ≤ 7 4 x 1 + 2 x 2 + x 3 = 1 2 .

Write the linear inequality constraints.

Write the linear equality constraints.

Write the bound constraints.

## Use Initial Point

Compare the number of steps to solve an integer programming problem both with and without an initial feasible point. The problem has eight variables, four linear equality constraints, and has all variables restricted to be positive.

Define the linear equality constraint matrix and vector.

Set lower bounds that restrict all variables to be nonnegative.

Specify that all variables are integer-valued.

Set the objective function vector f .

Solve the problem without using an initial point, and examine the display to see the number of branch-and-bound nodes.

For comparison, find the solution using an initial feasible point.

Without an initial point, intlinprog took about 30,000 branch-and-bound steps.

Using an initial point, intlinprog took about 5,000 steps.

Giving an initial point does not always help. For this problem, giving an initial point saves time and computational steps. However, for some problems, giving an initial point can cause intlinprog to take more steps.

## Solve an MILP with Nondefault Options

min x ( - 3 x 1 - 2 x 2 - x 3 ) s u b j e c t t o { x 3 b i n a r y x 1 , x 2 ≥ 0 x 1 + x 2 + x 3 ≤ 7 4 x 1 + 2 x 2 + x 3 = 1 2

without showing iterative display.

Specify the solver inputs.

Specify no display.

Run the solver.

## Solve MILP Using Problem-Based Approach

This example shows how to set up a problem using the problem-based approach and then solve it using the solver-based approach. The problem is

Create an OptimizationProblem object named prob to represent this problem. To specify a binary variable, create an optimization variable with integer type, a lower bound of 0, and an upper bound of 1.

Convert the problem object to a problem structure.

Solve the resulting problem structure.

Both sol(1) and sol(3) are binary-valued. Which value corresponds to the binary optimization variable xb ?

The variable xb appears last in the Variables display, so xb corresponds to sol(3) = 1 . See Algorithms .

## Examine the MILP Solution and Process

Call intlinprog with more outputs to see solution details and process.

The goal is to solve the problem

Call intlinprog with all outputs.

The output structure shows numnodes is 0 . This means intlinprog solved the problem before branching. This is one indication that the result is reliable. Also, the absolutegap and relativegap fields are 0 . This is another indication that the result is reliable.

## Input Arguments

F — coefficient vector real vector | real array.

Coefficient vector, specified as a real vector or real array. The coefficient vector represents the objective function f'*x . The notation assumes that f is a column vector, but you are free to use a row vector or array. Internally, linprog converts f to the column vector f(:) .

If you specify f = [] , intlinprog tries to find a feasible point without trying to minimize an objective function.

Example: f = [4;2;-1.7];

Data Types: double

## intcon — Vector of integer constraints vector of integers

Vector of integer constraints, specified as a vector of positive integers. The values in intcon indicate the components of the decision variable x that are integer-valued. intcon has values from 1 through numel(f) .

intcon can also be an array. Internally, intlinprog converts an array intcon to the vector intcon(:) .

Example: intcon = [1,2,7] means x(1) , x(2) , and x(7) take only integer values.

## A — Linear inequality constraints real matrix

Linear inequality constraints, specified as a real matrix. A is an M -by- N matrix, where M is the number of inequalities, and N is the number of variables (length of f ). For large problems, pass A as a sparse matrix.

A encodes the M linear inequalities

A*x <= b ,

where x is the column vector of N variables x(:) , and b is a column vector with M elements.

For example, consider these inequalities:

x 1 + 2 x 2 ≤ 10 3 x 1 + 4 x 2 ≤ 20 5 x 1 + 6 x 2 ≤ 30.

Specify the inequalities by entering the following constraints.

Example: To specify that the x-components add up to 1 or less, take A = ones(1,N) and b = 1 .

## b — Linear inequality constraints real vector

Linear inequality constraints, specified as a real vector. b is an M -element vector related to the A matrix. If you pass b as a row vector, solvers internally convert b to the column vector b(:) . For large problems, pass b as a sparse vector.

b encodes the M linear inequalities

where x is the column vector of N variables x(:) , and A is a matrix of size M -by- N .

Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1 .

## Aeq — Linear equality constraints real matrix

Linear equality constraints, specified as a real matrix. Aeq is an Me -by- N matrix, where Me is the number of equalities, and N is the number of variables (length of f ). For large problems, pass Aeq as a sparse matrix.

Aeq encodes the Me linear equalities

Aeq*x = beq ,

where x is the column vector of N variables x(:) , and beq is a column vector with Me elements.

For example, consider these equalities:

x 1 + 2 x 2 + 3 x 3 = 10 2 x 1 + 4 x 2 + x 3 = 20.

Specify the equalities by entering the following constraints.

Example: To specify that the x-components sum to 1, take Aeq = ones(1,N) and beq = 1 .

## beq — Linear equality constraints real vector

Linear equality constraints, specified as a real vector. beq is an Me -element vector related to the Aeq matrix. If you pass beq as a row vector, solvers internally convert beq to the column vector beq(:) . For large problems, pass beq as a sparse vector.

beq encodes the Me linear equalities

where x is the column vector of N variables x(:) , and Aeq is a matrix of size Me -by- N .

Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1 .

## lb — Lower bounds [] (default) | real vector or array

Lower bounds, specified as a vector or array of doubles. lb represents the lower bounds elementwise in lb  ≤  x  ≤  ub .

Internally, intlinprog converts an array lb to the vector lb(:) .

Example: lb = [0;-Inf;4] means x(1) ≥ 0 , x(3) ≥ 4 .

## ub — Upper bounds [] (default) | real vector or array

Upper bounds, specified as a vector or array of doubles. ub represents the upper bounds elementwise in lb  ≤  x  ≤  ub .

Internally, intlinprog converts an array ub to the vector ub(:) .

Example: ub = [Inf;4;10] means x(2) ≤ 4 , x(3) ≤ 10 .

## x0 — Initial point [] (default) | real array

Initial point, specified as a real array. The number of elements in x0 is the same as the number of elements of f , when f exists. Otherwise, the number is the same as the number of columns of A or Aeq . Internally, the solver converts an array x0 into a vector x0(:) .

Providing x0 can change the amount of time intlinprog takes to converge. It is difficult to predict how x0 affects the solver. For suggestions on using appropriate Heuristics with x0 , see Tips .

x0 must be feasible with respect to all constraints. If x0 is not feasible, the solver errors. If you do not have a feasible x0 , set x0 = [] .

Example: x0 = 100*rand(size(f))

## options — Options for intlinprog options created using optimoptions

Options for intlinprog , specified as the output of optimoptions .

Some options are absent from the optimoptions display. These options appear in italics in the following table. For details, see View Optimization Options .

Example: options = optimoptions('intlinprog','MaxTime',120)

## problem — Structure encapsulating inputs and options structure

Structure encapsulating the inputs and options, specified with the following fields.

You must specify at least these fields in the problem structure. Other fields are optional:

Example: problem.f = [1,2,3]; problem.intcon = [2,3]; problem.options = optimoptions('intlinprog'); problem.Aineq = [-3,-2,-1]; problem.bineq = -20; problem.lb = [-6.1,-1.2,7.3]; problem.solver = 'intlinprog';

Data Types: struct

## Output Arguments

X — solution real vector.

Solution, returned as a vector that minimizes f'*x subject to all bounds, integer constraints, and linear constraints.

When a problem is infeasible or unbounded, x is [] .

## fval — Objective value real scalar

Objective value, returned as the scalar value f'*x at the solution x .

When a problem is infeasible or unbounded, fval is [] .

## exitflag — Algorithm stopping condition integer

Algorithm stopping condition, returned as an integer identifying the reason the algorithm stopped. The following lists the values of exitflag and the corresponding reasons intlinprog stopped.

The exit message can give more detailed information on the reason intlinprog stopped, such as exceeding a tolerance.

Exitflags 3 and -9 relate to solutions that have large infeasibilities. These usually arise from linear constraint matrices that have large condition number, or problems that have large solution components. To correct these issues, try to scale the coefficient matrices, eliminate redundant linear constraints, or give tighter bounds on the variables.

## output — Solution process summary structure

Solution process summary, returned as a structure containing information about the optimization process.

## Limitations

Often, some supposedly integer-valued components of the solution x(intCon) are not precisely integers. intlinprog deems as integers all solution values within IntegerTolerance of an integer.

To round all supposed integers to be exactly integers, use the round function.

Rounding solutions can cause the solution to become infeasible. Check feasibility after rounding:

intlinprog does not enforce that solution components be integer-valued when their absolute values exceed 2.1e9 . When your solution has such components, intlinprog warns you. If you receive this warning, check the solution to see whether supposedly integer-valued components of the solution are close to integers.

intlinprog does not allow components of the problem, such as coefficients in f , A , or ub , to exceed 1e25 in absolute value. If you try to run intlinprog with such a problem, intlinprog issues an error.

To specify binary variables, set the variables to be integers in intcon , and give them lower bounds of 0 and upper bounds of 1 .

Save memory by specifying sparse linear constraint matrices A and Aeq . However, you cannot use sparse matrices for b and beq .

If you include an x0 argument, intlinprog uses that value in the 'rins' and guided diving heuristics until it finds a better integer-feasible point. So when you provide x0 , you can obtain good results by setting the 'Heuristics' option to 'rins-diving' or another setting that uses 'rins' .

To provide logical indices for integer components, meaning a binary vector with 1 indicating an integer, convert to intcon form using find . For example,

intlinprog replaces bintprog . To update old bintprog code to use intlinprog , make the following changes:

Set intcon to 1:numVars , where numVars is the number of variables in your problem.

Set lb to zeros(numVars,1) .

Set ub to ones(numVars,1) .

Update any relevant options. Use optimoptions to create options for intlinprog .

Change your call to bintprog as follows:

## Alternative Functionality

The Optimize Live Editor task provides a visual interface for intlinprog .

## Version History

R2019a: default branchrule is 'reliability'.

The default value of the BranchRule option is 'reliability' instead of 'maxpscost' . In testing, this value gave better performance on many problems, both in solution times and in number of explored branching nodes.

On a few problems, the previous branch rule performs better. To get the previous behavior, set the BranchRule option to 'maxpscost' .

linprog | mpsread | optimoptions | prob2struct | Optimize

• Mixed-Integer Linear Programming Basics: Solver-Based
• Factory, Warehouse, Sales Allocation Model: Solver-Based
• Traveling Salesman Problem: Solver-Based
• Solve Sudoku Puzzles via Integer Programming: Solver-Based
• Mixed-Integer Quadratic Programming Portfolio Optimization: Solver-Based
• Optimal Dispatch of Power Generators: Solver-Based
• Mixed-Integer Linear Programming (MILP) Algorithms
• Tuning Integer Linear Programming
• Solver-Based Optimization Problem Setup

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Solver of multiobjective linear optimization problems: description and documents

View the Project on GitHub vOptSolver/vOptSolver

## Solver of multiobjective linear optimization problems

vOptSolver is an ecosystem for modeling and solving multiobjective linear optimization problems (MOMIP, MOLP, MOIP, MOCO). It integrates several exact algorithms for computing the set of non-dominated points $Y_N$, and the corresponding complete set of efficient solutions $X_E$, for structured and non-structured optimization problems with at least two objectives.

It is composed of the two julia packages MultiObjectiveAlgorithms (previously vOptGeneric ) and vOptSpecific , and hosts vOPtLib , a library of numerical instances:

• MultiObjectiveAlgorithms.jl which replaces vOptGeneric since February 2023
• vOptSpecific

IMPORTANT (Feb-2023): vOptGeneric.jl has been fully redesigned, and reimplemented. It becomes MultiObjectiveAlgorithms.jl (MOA) , a collection of algorithms for multi-objective optimization integrated to JuMP and MathOptInterface . MOA comes with an enriched list of multi-objective algorithms, especially for solving problems with 3 objectives. Consequently vOptGeneric.jl is no longer under active development. It will remain available on Github at https://github.com/vOptSolver/vOptGeneric.jl. From February 2023, the JuMP-dev organization will continue to maintain the MOA package and transition development over the long term.

## Instructions

• Documentation

All bugs, feature requests, pull requests, feedback, etc., are welcome.

## Coordinator

Prof. Dr. Xavier Gandibleux, University of Nantes - France (contact)

By alphabetical order (concerning vOptGeneric and vOptSpecific):

• Current contributors: Xavier Gandibleux, Anthony Przybylski, Gauthier Soleilhac.
• Past contributors: Lucas Baussey, Pauline Chatelier, Quentin Delmée, Dorian Dumez, Flavien Lucas, Clément Turcat.

## How To Contribute

• in adding your examples (code JuMP + data) solved with vOptGeneric to the collection;
• in plugging your own C/C++/Julia algorithms into vOptSpecific or vOptGeneric;
• in adapting vOptSpecific for windows;
• in sending us your suggestions to improve/extend vOptSolver;
• in telling us when you have completed a work (exercices for students; research; paper; etc.) using vOptSolver;
• in joining the adventure with us as maintainer of the solver, repositories, documents, etc.

In brief, every contributions aiming to share our efforts, our algorithms, our productions around this open source software are welcome.

Xavier Gandibleux. Multi-objective optimization with JuMP. JuliaCon 2023 . Massachusetts Institute of Technology. Cambridge, USA. July 25-29, 2023.

Xavier Gandibleux, Gauthier Soleilhac, Anthony Przybylski. vOptSolver: an ecosystem for multi-objective linear optimization. JuliaCon 2021 . Online and everywhere. July 28-30, 2021. Abstract . Reference to use for citing vOptSolver

Anthony Przybylski. Optimisation combinatoire multi-objectif : méthodes de résolution exacte et solveur vOpt. JuliaDay’2019 : Journée « Julia et Optimisation ». 17 Juin 2019 . Université de Nantes, France. https://julialang.univ-nantes.fr/journee-julia-et-optimisation

Xavier Gandibleux et Anthony Przybylski, Algorithmes de branch-and-bound multiobjectif et vOptSolver. Tutoriel du GDR CNRS RO, ROADEF’2018 : 19e édition du congrès annuel de la Société Française de Recherche Opérationnelle et d’Aide à la Décision. 22 février 2018, Lorient. Video .

Xavier Gandibleux et Anthony Przybylski, Optimisation combinatoire multiobjectif : deux contributions issues du projet de recherche ANR-DFG vOpt. SPOC16 : 16e journée “Polyèdres et optimisation combinatoire” . LAMSADE/Université Paris-Dauphine. 15 décembre 2017.

Xavier Gandibleux, Pascal Halffmann. Designing and Experimenting with vOptSolver an Algorithm for Computing the Weight Set Decomposition. RAMOO’2017 : 4th International workshop “Recent Advances in Multi-Objective Optimization” , TU-Kaiserslautern, Germany. 2017.

Xavier Gandibleux, Gauthier Soleilhac, Anthony Przybylski, Flavien Lucas, Stefan Ruzika, Pascal Halffmann. vOptSolver, a “get and run” solver of multiobjective linear optimization problems built on Julia and JuMP. MCDM2017: 24th International Conference on Multiple Criteria Decision Making . July 10-14, 2017. Ottawa (Canada).

Xavier Gandibleux, Gauthier Soleilhac, Anthony Przybylski, Stefan Ruzika. vOptSolver: an open source software environment for multiobjective mathematical optimization. IFORS2017: 21st Conference of the International Federation of Operational Research Societies . July 17-21, 2017. Quebec City (Canada).

The development of vOptSolver started in the ANR/DFG-14-CE35-0034-01 research project vOpt (2015-2019) (link) involving Université de Nantes (France) and University of Koblenz-Landau/University of Kaiserslautern (Germany).

• Solver of multiobjective linear optimization problems for scientifics and practionners
• Easy to formulate a problem, to provide data, to solve a problem, to collect the outputs, to analyze the solutions
• Natural and intuitive use for mathematicians, informaticians, engineers
• Solving needs: methods and algorithms for performing numerical experiments
• Research needs: support and primitives for the development of new algorithms
• Pedagogic needs: environment for practicing of theories and algorithms

## Characteristics

• Efficient, flexible, evolutive solver
• Free, open source, multi-platform, reusing existing specifications
• Easy installation, no need of being expert in computer science
• Usual free (GLPK, Clp/Cbc) and commercial (CPLEX, GUROBI) MIP solvers

## Problems managed

• p-LP: Linear Program
• p-MIP: Mixed Integer linear Program
• p-IP: Integer linear Program
• 2-LAP: Linear Assignment Problem
• 2-OSP: One machine Scheduling Problem
• 2-UKP: binary Unidimensional knapsack problem
• 2-UMFLP: Uncapacitated Mixed variables Facility Location Problem
• Forthcoming: [p-PATHS, 2-UDFLP]. Projects : [MKP, SSCFLP, CFLP]

## Algorithms integrated

The solving algorithms included compute exact solution(s) corresponding to Y_{lex}, Y_{SN}, or Y_{N}.

• Lexico: compute Y_{lex}, the lexicographic optimal solutions for p-IP (Julia+JuMP)
• Aneja1979: compute Y_{SN} with Aneja & Nair method (also named the dichotomic method) for 2-IP (Julia+JuMP)
• Haimes1971: compute Y_{N} with epsilon-constraint method for 2-IP (Julia+JuMP)
• Chalmet1986: compute Y_N with Chalmet et al. method for 2-IP (Julia+JuMP)
• Forthcoming: algorithms for [{2,3}-LP]. Projects: algorithms for [3-IP, {2,3}-MIP]
• Przybylski2008: 2LAP2008 (C)
• Wassenhove1980: 2OSP1980 (implemented in 2017 in Julia)
• Jorge2010: 2UKP2010 (re-implemented in 2017 in Julia)
• Delmee2017: 2UMFLP2016 (C++)
• Forthcoming:[Gandibleux2006: PATHS (re-implemented in 2018 in Julia); Gandibleux2012: 2UDFLP2012 (re-implemented in 2018 in C)]
• direct with the provided languages (Julia, JuMP)
• standard MOP format (IP, MIP, LP)
• specific problem format (MIP)
• direct with the language (Julia),
• specific problem format (2LAP, 2UKP, 2UFLP)
• standard 2MOP format (IP, MIP, LP)

## Information

• Julia is available on macOS, linux, and windows
• vOptSolver (composed of vOptGeneric and vOptSpecific) is free, open source under MIT licence.
• vOptGeneric has been tested with Julia 1.6 on macOS 11.4, linux-Ubuntu 18.04.5 LTS and Windows 10 64 bits.
• vOptSpecific have been tested with Julia 1.6 on macOS 11.4 and linux-Ubuntu 18.04.5 LTS.

## Installation and usage Instructions

Refer to the instructions provided for

• vOptGeneric

## Documentation and examples

NB: the available documentation is obsolete (written for Julia v0.6.4; new documentation compliant with v1.x is coming).

Old documentation:

• Tutorial (in waiting a user manual) for new users (folder docs)
• Presentations given in conferences (folder talks)

Examples of problems ready to be solved:

• With vOptGeneric (folder examples)
• With vOptSpecific (folder examples)

[Haimes1971] Y.V. Haimes, L.S. Lasdon, D.A. Wismer: On a bicriterion formation of the problems of integrated system identification and system optimization. IEEE Transactions on Systems, Man and Cybernetics , Volume SMC-1, Issue 3, Pages 296-297, July 1971.

[Aneja1979] Y. P. Aneja and K. P. K. Nair: Bicriteria Transportation Problem. Management Science , 25:1, 73-78 1979.

[Wassenhove1980] L. N. Van Wassenhove, L. F. Gelders: Solving a bicriterion scheduling problem. European Journal of Operational Research , Volume 4, Issue 1, Pages 42-48, 1980.

[Chalmet1986] L.G. Chalmet, L. Lemonidis, D.J. Elzinga: An algorithm for the bi-criterion integer programming problem. European Journal of Operational Research , Volume 25, Issue 2, Pages 292-300, 1986.

[Gandibleux2006] X. Gandibleux, F. Beugnies, S. Randriamasy: Martins’ algorithm revisited for multi-objective shortest path problems with a MaxMin cost function. 4OR: A Quarterly Journal of Operations Research , Springer Verlag, 4 (1), pp.47-59, 2006.

[Przybylski2008] A. Przybylski, X. Gandibleux, M. Ehrgott: Two phase algorithms for the bi-objective assignment problem. European Journal of Operational Research , Volume 185, Issue 2, Pages 509-533, 2008.

[Jorge2010] J. Jorge: Nouvelles propositions pour la résolution exacte du sac à dos multi-objectif unidimensionnel en variables binaires. PhD Thesis (in French), Université de Nantes - France, 2010.

[Gandibleux2012] X. Gandibleux, A. Przybylski , S. Bourougaa, A. Derrien, A. Grimault: Computing the Efficient Frontier for the 0/1 Biobjective Uncapacitated Facility Location Problem CORS/MOPGP’2012 (10th international conference on Multiple Objective Programming and Goal Programming). June 11-13, 2012, Niagara Falls, Canada.

[Vincent2013] Th. Vincent: Caractérisation des solutions efficaces et algorithmes d’énumération exacts pour l’optimisation multiobjectif en variables mixtes binaires. PhD Thesis (in French), Université de Nantes - France, 2013.

[Delmee2017] Q. Delmée, X. Gandibleux, A. Przybylski: Résolution exacte du problème de localisation de services bi-objectif sans contrainte de capacité en variables mixtes. ROADEF2017 (18ème édition du congrès annuel de la Société Française de Recherche Opérationnelle et d’Aide à la Décision). 22-24 février 2017, Metz, France.

[Dumez2017] D. Dumez, X. Gandibleux, I. Rusu. Datastructures for Filtering and Storing Non-Dominated Points. MOPGP’2017: 12th International Conference on Multiple Objective Programming and Goal Programming. 30-31 October 2017, Metz, France.

Terms and acronyms used

• LP: Linear Program
• MIP: Mixed Integer linear Program
• IP: Integer linear program
• CO: Combinatorial Optimization
• MOLP: MultiObjective linear program
• MOIP: MultiObjective Integer linear program
• MOMIP: MultiObjective Mixed Integer linear Program
• MOCO: MultiObjective Combinatorial Optimization
• OSP: One machine Scheduling Problem
• LAP: Linear Assignment Problem
• UKP: Unidimensional 01 Knapsack Problem
• MKP: Multidimensional 01 Knapsack Problem
• UFLP: Uncapacitated Facility Location Problem
• UDFLP: Discrete Uncapacitated Facility Location Problem
• SSCFLP: Single Source Capacitated Facility Location Problem
• UMFLP: Uncapacitated Mixed variables Facility Location Problem
• CFLP: Capacitated Facility Location Problem
• PATHS: shortest paths problem
• Julia: name of the programming language
• JuMP: stands for Julia for Mathematical Optimization, a modeling language for mathematical optimization embedded in Julia
• AVL tree is a self-balancing binary search tree
• API: stands for Application Programming Interface
• GPL: stands for GNU General Public License
• GLPK: stands for GNU Linear Programming Kit, an open source solver
• Clp/Cbc : an open source solver (for LP and MILP respectively) from the COIN-OR project
• CPLEX: a commercial solver
• GUROBI: a commercial solver
• MOP: MultiObjective extension of MPS format
• Calculators
• Basic Calculators

## Linear Programming Calculator

Linear Programming Calculator is a free online tool that displays the best optimal solution for the given constraints. BYJU’S online linear programming calculator tool makes the calculations faster, and it displays the best optimal solution for the given objective functions with the system of linear constraints in a fraction of seconds.

## How to Use Linear Programming Calculator?

The procedure to use the linear programming calculator is as follows: Step 1: Enter the objective function, constraints in the respective input field Step 2: Now click the button “Submit” to get the optimal solution Step 3: Finally, the best optimal solution and the graph will be displayed in the new window

## What is Linear Programming?

Linear programming is the best optimization technique which gives the optimal solution for the given objective function with the system of linear constraints. The main goal of this technique is finding the variable values that maximise or minimize the given objective function. Here, the objective function defines the amount to be optimised, and the constraints define the range. The four main components of linear programming are:

• Objective Function
• Constraints
• Decision Variables

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## Integer Linear Programming

Integer Linear Programming (ILP) is a type of optimization problem where the variables are integer values and the objective function and equations are linear.

\begin{align} & \text{maximize} && \mathbf{c}^\mathrm{T} \mathbf{x}\\ & \text{subject to} && A \mathbf{x} \le \mathbf{b} \\ & && \mathbf{x} \ge \mathbf{0} \\ &&& \mathbf{x} \in \mathbb{Z}^n \end{align}

A Mixed-Integer Linear Programming (MILP) problem has continuous and integer variables. Mixed-Integer Nonlinear Programming (MINLP) also includes nonlinear equations and requires specialized MINLP solvers such as APOPT . MINLP solvers can also solve MILP or ILP problems although other solvers such as CPLEX, Gurobi, or FICO Xpress are specialized commercial solvers for MILP. APMonitor and GEKKO solve MINLP problems. The following integer linear programming (ILP) problem has a two potential maximum values at (1,2) and (2,2).

APMonitor Model File

Integer variables are declared in APMonitor with the prefix int . The problem is solved with MATLAB , Julia , Python or through the APMonitor Online Interface .

Python Gekko

Gekko is a Python API to APMonitor. The same ILP problem is solved with Gekko.

Nonlinear programming solvers (such as IPOPT) may not return an integer solution because they are designed for continuous variables. Mixed Integer Nonlinear Programming solvers (such as APOPT) are equipped to solve for binary or integer variables. It is selected with m.options.SOLVER=1 . Select the appropriate solver option to either find an initial solution without integer variables or an integer solution. It is sometimes desirable to find a non-integer solution first because of the often significant reduction in computation time without the integer variables.

Solution in Matrix Form

Another representation is matrix form. Gekko function qobj defines a linear or quadratic objective and axb defines the Ax\leb constraint.

Solution in Sparse Matrix Form

For large-scale problems, it is more efficient to solve the problem in sparse matrix form. The matrix arguments are passed to Gekko in coordinate (COO) list form. COO list form is [row indices],[values] for a vector and [row indices],[column indices],[values] for a matrix.

• Dynamic Optimization with Discrete Variables
• Discrete Optimization
• Linear Programming
• Mixed Integer Nonlinear Programming (MINLP)

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## Get started with OR-Tools

Learn how to solve optimization problems from C++, Python, C#, or Java.

## Install OR-Tools

See the Release Notes for the latest updates.

## OR-Tools won gold in the international constraint programming competition every year since 2013.

OR-Tools is an open source software suite for optimization, tuned for tackling the world's toughest problems in vehicle routing, flows, integer and linear programming, and constraint programming.

After modeling your problem in the programming language of your choice, you can use any of a half dozen solvers to solve it: commercial solvers such as Gurobi or CPLEX, or open-source solvers such as SCIP, GLPK, or Google's GLOP and award-winning CP-SAT.

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## Excel Solver - Integer Programming

• Excel Solver - What Solver Can and Cannot Do
• Excel Solver - Optimization Methods

When a Solver model includes integer, binary or alldifferent constraints, it is called an integer programming problem.  Integer constraints make a model non-convex , and finding the optimal solution to an integer programming problem is equivalent to solving a global optimization problem.  Such problems may require far more computing time than the same problem without the integer constraints.  When the Simplex LP or GRG Nonlinear Solving methods are used, Solver uses a Branch & Bound method for the integer constraints.  The Evolutionary Solving method uses its own methods for such problems.

The Branch & Bound Method

The Branch & Bound method begins by finding the optimal solution to the “relaxation” of the problem, ignoring the integer constraints.  If it happens that in this solution, the decision variables with integer constraints already have integer values, then no further work is required.  If one or more integer variables have non-integral solutions, the Branch & Bound method chooses one such variable and “branches,” creating two new subproblems where the value of that variable is more tightly constrained.  For example, if integer variable A1 has the value 3.45 at the solution, then one subproblem will have the additional constraint A1 <= 3 and the other subproblem will add the constraint A1 >= 4.  These subproblems are solved and the process is repeated, “branching” as needed on each of the integer decision variables, until a solution is found where all of the integer variables have integer values (to within a small tolerance).

Hence, the Branch & Bound method may solve many subproblems, each one a “regular” Solver problem.  The number of subproblems may grow exponentially .  The “bounding” part of the Branch & Bound method is designed to eliminate sets of subproblems that do not need to be explored because the resulting solutions cannot be better than the solutions already obtained.

Many other methods , such as preprocessing and probing, cut generation, node and branch variable selection, local search and heuristics, can be applied to integer programming problems to mitigate the exponential growth of the number of subproblems, but all these methods normally operate within the Branch & Bound framework.

Optimization Online

## The SCIP Optimization Suite 4.0

• Tobias Fischer
• Tristan Gally
• Gerald Gamrath
• Ambros Gleixner
• Robert Lion Gottwald
• Gregor Hendel
• Thorsten Koch
• Marco E. Lübbecke
• Stephen J. Maher
• Matthias Miltenberger
• Marc E. Pfetsch
• Felipe Serrano
• Dieter Weninger
• Jakob Witzig
• Benjamin Müller
• Christian Puchert
• Daniel Rehfeldt
• Sebastian Schenker
• Robert Schwarz
• Yuji Shinano
• Jonas T. Witt

The SCIP Optimization Suite is a powerful collection of optimization software that consists of the branch-cut-and-price framework and mixed-integer programming solver SCIP, the linear programming solver SoPlex, the modeling language Zimpl, the parallelization framework UG, and the generic branch-cut-and-price solver GCG. Additionally, it features the extensions SCIP-Jack for solving Steiner tree problems, PolySCIP for solving multi-objective problems, and SCIP-SDP for solving mixed-integer semidefinite programs. The SCIP Optimization Suite has been continuously developed and has now reached version 4.0. The goal of this report is to present the recent changes to the collection. We not only describe the theoretical basis, but focus on implementation aspects and their computational consequences.

ZIB-Report 17-12, Zuse Institute Berlin, Takustr. 7, 14195 Berlin, March 2017

View The SCIP Optimization Suite 4.0

#### IMAGES

1. How to Solve Integer Linear Programming in Excel (With Easy Steps)

2. Integer Linear Programming: Excel Solver Example 1

3. How to solve an Integer Linear Programming Problem Using Branch and Bound

4. How to Solve Linear Programming (LP) Problems Using Python

5. How to Use Excel Solver for Linear Programming (with Easy Steps)

6. Integer Linear Programming

#### VIDEO

1. Vid03 Problem formulation and Integer Linear Programming

2. Linear Programming

3. Linear Programming Problem(Part 1)

4. Linear Programming Classic Problems 10

5. Integer Linear programming/Module 2 part 1 /Optimization Techniques/MGU

6. Linear Programming

1. Linear Programming Solver

Linear Programming Solver Added Jul 31, 2018 by vik_31415 in Mathematics Linear programming solver with up to 9 variables. New constraints could be added by using commas to separate them. Send feedback | Visit Wolfram|Alpha Get the free "Linear Programming Solver" widget for your website, blog, Wordpress, Blogger, or iGoogle.

2. Linear Optimization

Online Linear and Integer Optimization Solver

3. Graphical Method Calculator

Solving optimization exercises with the graphical method will be easier with our graphical method calculator for linear programming problems. Graphical Method Calculator - Linear Programming Objective: Objective Function: X + X Constraints Constraint 1: X + X Constraint 2: X + X X 1, X 2 ≥ 0 + − Graph Reset Members-Only Content

4. Branch and Bound method calculator

Solution provided by AtoZmath.com. Branch and Bound method calculator. subject to x2 ≤ 7 2. Solve the following LP problem by using Branch and Bound method Max Z = 3x1 + 5x2 subject to 2x1 + 4x2 ≤ 25 x1 ≤ 8 2x2 ≤ 10 and x1,x2 ≥ 0. Share this solution or page with your friends.

5. SCIP

very fast standalone solver for linear programming (LP), mixed integer programming (MIP), and mixed integer nonlinear programming (MINLP) framework for branching, cutting plane separation, propagation, pricing, and Benders' decomposition, highly flexible through many possible user plugins: constraint handlers to implement arbitrary constraints,

6. linear programming calculator

Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step

7. Simplex method tool

Use of this system is pretty intuitive. Press the "Examples" buttons to see how to format your linear programming problem. Notes. This page also solves integer and mixed linear programing problems (using the "branch and bound" algorithm). To specify which variables are required to have integer values, use the format integer x,y,z (see the ...

8. Linear Programming Calculator + Online Solver With Free Steps

Linear Programming Calculator is a free online calculator that provides the best optimal solution for the given mathematical model. This online calculator solves the problem of finding the correct solution or optimized output of the desired mathematical models by providing a quick, reliable, and accurate solution.

9. Exploring Mixed-Integer and Linear Programming Solvers

Common Free Mixed-Integer and Linear Programming Solvers There are several open-source packages available for optimization. Two popular ones are GLPK and LP_Solve. The nice thing about these packages is that it's easy to get started because you can download and install them immediately.

10. Mixed-integer linear programming (MILP)

Mixed-integer linear programming solver. Finds the minimum of a problem specified by min x f T x subject to { x ( intcon) are integers A ⋅ x ≤ b A e q ⋅ x = b e q l b ≤ x ≤ u b. f, x, intcon, b, beq, lb , and ub are vectors, and A and Aeq are matrices. You can specify f, intcon, lb , and ub as vectors or arrays. See Matrix Arguments. Note

11. NEOS Solvers

DSP [ SMPS ] The NEOS Server optimization solvers represent the state-of-the-art in computational optimization. Optimization problems are solved automatically with minimal input from the user. Users only need a definition of the optimization problem; all additional information required by the optimization solver is determined automatically.

12. Solver of multiobjective linear optimization problems

p-MIP: Mixed Integer linear Program; p-IP: Integer linear Program; vOptSpecific: Multiobjective structured problems / Application Programming Interface (API), ... GLPK: stands for GNU Linear Programming Kit, an open source solver; Clp/Cbc : an open source solver (for LP and MILP respectively) from the COIN-OR project; CPLEX: a commercial solver;

13. Linear Programming Calculator

Linear Programming Calculator is a free online tool that displays the best optimal solution for the given constraints. BYJU'S online linear programming calculator tool makes the calculations faster, and it displays the best optimal solution for the given objective functions with the system of linear constraints in a fraction of seconds.

14. OR-Tools

Linear optimization problems that require some of the variables to be integers are called Mixed Integer Programs (MIPs). These variables can arise in a couple of ways: Integer variables that...

15. Linear Programming

The elements of the Q column are calculated by dividing the values from column P by the value from the column corresponding to the variable that is entered in the basis: Q 1 = P 1 / x 1,6 = 245 / -0.3 = -816.67; Q 2 = P 2 / x 2,6 = 225 / 0 = ∞; Q 3 = P 3 / x 3,6 = 140 / 0.4 = 350;

16. Integer Linear Programming

Integer Linear Programming (ILP) is a type of optimization problem where the variables are integer values and the objective function and equations are linear. maximize cT x subject to Ax≤ b x ≥0 x ∈Zn maximize c T x subject to A x ≤ b x ≥ 0 x ∈ Z n A Mixed-Integer Linear Programming (MILP) problem has continuous and integer variables.

17. How to choose an integer linear programming solver?

5 Answers. If what you want is linear mixed integer programming, then I would point to Coin-OR (and specifically to the module CBC). It's Free software (as speech) You can either use it with a specific language, or use C++. Use C++ if you data requires lots of preprocessing, or if you want to put your hands into the solver (choosing pivot ...

18. Linear Programming Project Graph

Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Linear Programming Project Graph | Desmos

19. OR-Tools

OR-Tools is an open source software suite for optimization, tuned for tackling the world's toughest problems in vehicle routing, flows, integer and linear programming, and constraint...

20. linear programming

In the Mixed Integer Linear Programming Benchmark with 12 threads and a time limit of 2 hours MIPCL managed to solve 79 instances. Only the commercial solvers CPLEX, Gurobi and XPRESS managed to solve more under the given constraints (86 or 87 instances, respectively). Also in terms of the chosen performance metric (again using 12 threads ...

21. Best Open-Source Mixed Integer Optimization Solver

We focus on three commonly used free and open-source MIO solvers: GLPK (GNU linear programming kit) is capable of solving large-scale linear, integer, mixed-integer, and related problems. GLPK uses the simplex method for solving linear problems and branch-and-bound for dealing with integer optimization problems.

22. Excel Solver

These subproblems are solved and the process is repeated, "branching" as needed on each of the integer decision variables, until a solution is found where all of the integer variables have integer values (to within a small tolerance). Hence, the Branch & Bound method may solve many subproblems, each one a "regular" Solver problem.

23. The SCIP Optimization Suite 4.0

The SCIP Optimization Suite 4.0. The SCIP Optimization Suite is a powerful collection of optimization software that consists of the branch-cut-and-price framework and mixed-integer programming solver SCIP, the linear programming solver SoPlex, the modeling language Zimpl, the parallelization framework UG, and the generic branch-cut-and-price ...