Seconds For Problems Research Articles

Solving large break minimization problems in a mirrored double round-robin tournament using quantum annealing.

Quantum annealing has gained considerable attention because it can be applied to combinatorial optimization problems, which have numerous applications in logistics, scheduling, and finance. In recent years, with the technical development of quantum annealers, research on solving practical combinatorial optimization problems using them has accelerated. However, researchers struggle to find practical combinatorial optimization problems, for which quantum annealers outperform mathematical optimization solvers. Moreover, there are only a few studies that compare the performance of quantum annealers with the state-of-the-art solvers, such as Gurobi and CPLEX. This study determines that quantum annealing demonstrates better performance than the solvers in that the solvers take longer to reach the objective function value of the solution obtained by the quantum annealers for the break minimization problem in a mirrored double round-robin tournament. We also explain the desirable performance of quantum annealing for the sparse interaction between variables and a problem without constraints. In this process, we demonstrate that this problem can be expressed as a 4-regular graph. Through computational experiments, we solve this problem using our quantum annealing approach and two-integer programming approaches, which were performed using the latest quantum annealer D-Wave Advantage, and Gurobi, respectively. Further, we compare the quality of the solutions and the computational time. Quantum annealing was able to determine the exact solution in 0.05 seconds for problems with 20 teams, which is a practical size. In the case of 36 teams, it took 84.8 s for the integer programming method to reach the objective function value, which was obtained by the quantum annealer in 0.05 s. These results not only present the break minimization problem in a mirrored double round-robin tournament as an example of applying quantum annealing to practical optimization problems, but also contribute to find problems that can be effectively solved by quantum annealing.

Solving assembly scheduling problems with tree-structure precedence constraints: A Lagrangian relaxation approach

In this paper, we consider an assembly scheduling problem (ASP) with tree-structured precedence constraints. In our problem, there are a number of work centers. Each work center contains a number of machines of the same functionality. The job to be processed via this system is a job with tree-structure precedence constraints. Each operation in the job has a designated work center. We propose a mixed integer linear programming formulation and solve the problem with a Lagrangian relaxation (LR) approach. We solve the subproblems of the LR problem via a heuristic method and generate feasible solutions via a randomized list scheduling algorithm. Near-optimal results are obtained and the computational time is within a few seconds for problems with size up to 20 machines and 300 operations.

A new version of Scilab software package for the study of dynamical systems

This work presents a new version of a software package for the study of chaotic flows, maps and fractals [1]. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov–Sinai entropy. Various well-known examples are implemented, with the capability of the users inserting their own ODE or iterative equations. New version program summaryProgram title: Chaos v2.0Catalogue identifier: AEAP_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAP_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1275No. of bytes in distributed program, including test data, etc.: 7135Distribution format: tar.gzProgramming language: Scilab 5.1.1. Scilab 5.1.1 should be installed before running the program. Information about the installation can be found at http://wiki.scilab.org/howto/install/windows.Computer: PC-compatible running Scilab on MS Windows or LinuxOperating system: Windows XP, LinuxRAM: below 150 MegabytesClassification: 6.2Catalogue identifier of previous version: AEAP_v1_0Journal reference of previous version: Comput. Phys. Comm. 178 (2008) 788Does the new version supersede the previous version?: YesNature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE).Solution method:1.Numerical solving of ordinary differential equations for the study of chaotic flows. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincare sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov–Sinai entropies.2.Numerical solving of iterative equations for the study of maps and fractals.Reasons for new version: The program has been updated to use the new version 5.1.1 of Scilab with new graphical capabilities [2]. Moreover, new use cases have been added which make the handling of the program easier and more efficient.Summary of revisions:1.A new use case concerning coupled predator-prey models has been added [3].2.Three new use cases concerning fractals (Sierpinsky gasket, Barnsley's Fern and Tree) have been added [3].3.The graphical user interface (GUI) of the program has been reconstructed to include the new use cases.4.The program has been updated to use Scilab 5.1.1 with the new graphical capabilities.Additional comments: The program package contains 12 subprograms.•interface.sce – the graphical user interface (GUI) that permits the choice of a routine as follows•1.sci – Lorenz dynamical system•2.sci – Chua dynamical system•3.sci – Rosler dynamical system•4.sci – Henon map•5.sci – Lyapunov exponents for Lorenz dynamical system•6.sci – Lyapunov exponent for the logistic map•7.sci – Shannon entropy for the logistic map•8.sci – Coupled predator-prey model•1f.sci – Sierpinsky gasket•2f.sci – Barnsley's Fern•3f.sci – Barnsley's TreeRunning time: 10 to 20 seconds for problems that do not involve Lyapunov exponents calculation; 60 to 1000 seconds for problems that involve high orders ODE, Lyapunov exponents calculation and fractals.

Scilab software package for the study of dynamical systems

This work presents a new software package for the study of chaotic flows and maps. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov–Sinai entropy. Various well known examples are implemented, with the capability of the users inserting their own ODE. Program summary Program title: Chaos Catalogue identifier: AEAP_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAP_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 885 No. of bytes in distributed program, including test data, etc.: 5925 Distribution format: tar.gz Programming language: Scilab 3.1.1 Computer: PC-compatible running Scilab on MS Windows or Linux Operating system: Windows XP, Linux RAM: below 100 Megabytes Classification: 6.2 Nature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE). Solution method: Numerical solving of ordinary differential equations. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincaré sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov–Sinai entropies. Restrictions: The package routines are normally able to handle ODE systems of high orders (up to order twelve and possibly higher), depending on the nature of the problem. Running time: 10 to 20 seconds for problems that do not involve Lyapunov exponents calculation; 60 to 1000 seconds for problems that involve high orders ODE and Lyapunov exponents calculation.

GeM software package for computation of symmetries and conservation laws of differential equations

We present a recently developed Maple-based “GeM” software package for automated symmetry and conservation law analysis of systems of partial and ordinary differential equations (DE). The package contains a collection of powerful easy-to-use routines for mathematicians and applied researchers. A standard program that employs “GeM” routines for symmetry, adjoint symmetry or conservation law analysis of any given DE system occupies several lines of Maple code, and produces output in the canonical form. Classification of symmetries and conservation laws with respect to constitutive functions and parameters present in the given DE system is implemented. The “GeM” package is being successfully used in ongoing research. Run examples include classical and new results. Program summary Title of program: GeM Catalogue identifier: ADYK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADYK_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: none Computers: PC-compatible running Maple on MS Windows or Linux; SUN systems running Maple for Unix on OS Solaris Operating systems under which the program has been tested: Windows 2000, Windows XP, Linux, Solaris Programming language used: Maple 9.5 Memory required to execute with typical data: below 100 Megabytes No. of lines in distributed program, including test data, etc.: 4939 No. of bytes in distributed program, including test data, etc.: 166 906 Distribution format: tar.gz Nature of physical problem: Any physical model containing linear or nonlinear partial or ordinary differential equations. Method of solution: Symbolic computation of Lie, higher and approximate symmetries by Lie's algorithm. Symbolic computation of conservation laws and adjoint symmetries by using multipliers and Euler operator properties. High performance is achieved by using an efficient representation of the system under consideration and resulting symmetry/conservation law determining equations: all dependent variables and derivatives are represented as symbols rather than functions or expressions. Restrictions on the complexity of the problem: The GeM module routines are normally able to handle ODE/PDE systems of high orders (up to order seven and possibly higher), depending on the nature of the problem. Classification of symmetries/conservation laws with respect to one or more arbitrary constitutive functions of one or two arguments is normally accomplished successfully. Typical running time: 1–20 seconds for problems that do not involve classification; 5–1000 seconds for problems that involve classification, depending on complexity.

A branch and bound algorithm to minimize completion time variance on a single processor

In this paper we discuss a single machine scheduling problem with the objective of minimizing the variance of job completion times. The CTV problem has been proved to be NP hard (Oper. Res. Lett. 14 (1993) 49) and no polynomial time algorithm exists to find an optimal solution for CTV minimization on single machine. Hence enumerative techniques and heuristics are used to get optimal and near optimal solutions, respectively. We present a branch and bound algorithm and extend the same algorithm to generate epsilon optimal solutions for large sized problems (i.e., number of jobs >30). The algorithm has been computationally tested, with randomly generated problems involving up to 100 jobs, using a personal computer (PC) with a 64 MB RAM capacity. The computational time required for generating optimal solutions are in few seconds for problems with jobs between 25 and 30. The performance of the branch and bound algorithm is compared with the pseudo-polynomial algorithm (Oper. Res. 40 (1992) 1148) for small sized problems. For problems with greater number of jobs, the epsilon optimal solutions obtained using branch and bound algorithm are compared with results of simulated annealing (Single machine scheduling with some non-regular objectives, M.S. Thesis, IIT Madras, 1997), tabu search (Proceedings of Operations Management Conference, IIT Madras, 2000) and heuristic proposed by Manna and Prasad (Eur. J. Oper. Res. 114 (1999) 411). Scope and purpose Scheduling jobs in manufacturing systems considering non-regular measures of performance have gained importance in recent years. The objectives of minimizing completion time variance ( CTV problem) and minimizing squares of deviations of job completion times from a given common due-date ( MSD problem) are the two problems that use non-regular measures, and have attracted attention in single machine scheduling and flowshop scheduling when jobs require uniform treatment i.e., each entity spends approximately same time in the system or waits for service as every other entity. This problem is very much relevant when a manufacturing/service system places emphasis on Just-in-Time philosophy. The purpose of this paper is to present an algorithm for minimizing completion time variance for a set of jobs to be processed on a single machine. The proposed algorithm is computationally evaluated with randomly generated problems involving up to 100 jobs and results are reported.

Congressional anecdotes

This work presents a new version of a software package for the study of chaotic flows, maps and fractals [1]. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov–Sinai entropy. Various well-known examples are implemented, with the capability of the users inserting their own ODE or iterative equations.Program title: Chaos v2.0Catalogue identifier: AEAP_v2_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAP_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1275No. of bytes in distributed program, including test data, etc.: 7135Distribution format: tar.gzProgramming language: Scilab 5.1.1. Scilab 5.1.1 should be installed before running the program. Information about the installation can be found at http://wiki.scilab.org/howto/install/windows.Computer: PC-compatible running Scilab on MS Windows or LinuxOperating system: Windows XP, LinuxRAM: below 150 MegabytesClassification: 6.2Catalogue identifier of previous version: AEAP_v1_0Journal reference of previous version: Comput. Phys. Comm. 178 (2008) 788Does the new version supersede the previous version?: YesNature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE).Solution method:1.Numerical solving of ordinary differential equations for the study of chaotic flows. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincare sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov–Sinai entropies.2.Numerical solving of iterative equations for the study of maps and fractals.Reasons for new version: The program has been updated to use the new version 5.1.1 of Scilab with new graphical capabilities [2]. Moreover, new use cases have been added which make the handling of the program easier and more efficient.Summary of revisions:1.A new use case concerning coupled predator-prey models has been added [3].2.Three new use cases concerning fractals (Sierpinsky gasket, Barnsley's Fern and Tree) have been added [3].3.The graphical user interface (GUI) of the program has been reconstructed to include the new use cases.4.The program has been updated to use Scilab 5.1.1 with the new graphical capabilities.Additional comments: The program package contains 12 subprograms.•interface.sce – the graphical user interface (GUI) that permits the choice of a routine as follows•1.sci – Lorenz dynamical system•2.sci – Chua dynamical system•3.sci – Rosler dynamical system•4.sci – Henon map•5.sci – Lyapunov exponents for Lorenz dynamical system•6.sci – Lyapunov exponent for the logistic map•7.sci – Shannon entropy for the logistic map•8.sci – Coupled predator-prey model•1f.sci – Sierpinsky gasket•2f.sci – Barnsley's Fern•3f.sci – Barnsley's TreeRunning time: 10 to 20 seconds for problems that do not involve Lyapunov exponents calculation; 60 to 1000 seconds for problems that involve high orders ODE, Lyapunov exponents calculation and fractals.

A parallel algorithm for constrained concave quadratic global minimization

The global minimization of large-scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of both a concave part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A linear underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables. These results were obtained on a four processor CRAY2 using both sequential and parallel implementations of the algorithm. The average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0.001. For a relative tolerance of 0.1, the average computation time appears to increase only linearly with the number of linear variables.