General Algebraic Modeling System

[Emelia Lute]

Thegeneral algebraic modeling system (GAMS) is a high-level modeling system for mathematical optimization. GAMS is designed for modeling and solving linear, nonlinear, and mixed-integer optimization problems. The system is tailored for complex, large-scale modeling applications and allows the user to build large maintainable models that can be adapted to new situations. The system is available for use on various computer platforms. Models are portable from one platform to another.

GAMS was the first algebraic modeling language (AML)[2] and is formally similar to commonly used fourth-generation programming languages. GAMS contains an integrated development environment (IDE) and is connected to a group of third-party optimization solvers. Among these solvers are BARON, COIN-OR solvers, CONOPT, COPT Cardinal Optimizer, CPLEX, DICOPT, IPOPT, MOSEK, SNOPT, and XPRESS.

GAMS allows the users to implement a sort of hybrid algorithm combining different solvers. Models are described in concise, human-readable algebraic statements. GAMS is among the most popular input formats for the NEOS Server.[citation needed] Although initially designed for applications related to economics and management science, it has a community of users from various backgrounds of engineering and science.

The driving force behind the development of GAMS were the users of mathematical programming who believed in optimization as a powerful and elegant framework for solving real life problems in science and engineering. At the same time, these users were frustrated by high costs, skill requirements, and an overall low reliability of applying optimization tools. Most of the system's initiatives and support for new development arose in response to problems in the fields of economics, finance, and chemical engineering, since these disciplines view and understand the world as a mathematical program.

The idea of an algebraic approach to represent, manipulate, and solve large-scale mathematical models fused old and new paradigms into a consistent and computationally tractable system. Using generator matrices for linear programs revealed the importance of naming rows and columns in a consistent manner. The connection to the emerging relational data model became evident. Experience using traditional programming languages to manage those name spaces naturally lead one to think in terms of sets and tuples, and this led to the relational data model.

Combining multi-dimensional algebraic notation with the relational data model was the obvious answer. Compiler writing techniques were by now widespread, and languages like GAMS could be implemented relatively quickly. However, translating this rigorous mathematical representation into the algorithm-specific format required the computation of partial derivatives on very large systems. In the 1970s, TRW developed a system called PROSE that took the ideas of chemical engineers to compute point derivatives that were exact derivatives at a given point, and to embed them in a consistent, Fortran-style calculus modeling language. The resulting system allowed the user to use automatically generated exact first and second order derivatives. This was a pioneering system and an important demonstration of a concept. However, PROSE had a number of shortcomings: it could not handle large systems, problem representation was tied to an array-type data structure that required address calculations, and the system did not provide access to state-of-the art solution methods. From linear programming, GAMS learned that exploitation of sparsity was key to solving large problems. Thus, the final piece of the puzzle was the use of sparse data structures.

A transportation problem from George Dantzig is used to provide a sample GAMS model.[6] This model is part of the model library which contains many more complete GAMS models. This problem finds a least cost shipping schedule that meets requirements at markets and supplies at factories.

Modeling activities at the World Bank are highlighted and typified. Requirements for successful modeling applications in such a strategic planning environment are examined. The resulting development of a General Algebraic Modeling System (GAMS) is described. The data structure of this system is analyzed in some detail, and comparisons to other modeling systems are made. Selected aspects of the language are presented. The paper concludes with a case study of the Egyptian Fertilizer Sector in which GAMS has been used as a modeling tool.

Railway interlocking systems are crucial safety components in rail transportation, designed to prevent train collisions by regulating switch positions and signal indications. These systems delineate potential train movements within a railway station by connecting sections into routes, which are further divided into blocks. To ensure safety, the system prohibits the simultaneous allocation of the same block or intersecting routes to multiple trains. In this study, we characterize the 'interlocking problem' as a safety verification task for a single real-time station configuration, rather than a 'command and control' function. This is a matter of verification, not solution, typically managed by an interlocking system that receives movement authority requests. Over the years, we have developed various algebraic models to address this issue, suggesting the potential use of computer algebra systems in implementing interlocking systems. However, some of these models exhibit limitations. In this paper, we propose a novel algebraic model for decision-making in railway interlocking systems that overcomes the limitations of previous approaches, making it suitable for large railway stations. Our primary objective is to offer a mathematical solution to interlocking problems in linear time, which our approach accomplishes.

The General Algebraic Modeling System (GAMS release 2.25) is a software system designed for modeling linear, nonlinear and mixed integer optimization problems. The system, available on Strauss and Mahler, is especially useful for large complex models. GAMS allows you to model problems in a highly compact and natural way. After formulating the model, you can easily invoke several solvers to determine the solution to the problem. GAMS also provides an easy-to-use report writer to display the results of the solver in a useful format. At the University of Delaware, you can use any of three solvers:

The GAMS documentation describes approximately 100 models that serve as examples of simple and complex linear, nonlinear, and mixed integer formulations. These are stored online and can be copied and modified if you know either the name or the reference number of the model. For example, there is a blending problem known by the name "blend" and the reference number 47, in the GAMS printed documentation. To copy this model to your current working directory, type either of the following:

GAMS: General Algebraic Modeling System Linear and Nonlinear Programming. The full system documentation is provided electronically with the software and is also available on-line at: www.gams.com/docs/document.htm. The lectures are based on:

The lectures are based on: --McCarl, Bruce A., andThomas H. Spreen.Applied Mathematical Programming Using Algebraic Systems. Available at: --Paris, Q.An Economic Interpretation of Linear Programming.Iowa State University Press: Ames, Iowa, 1991.

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